BAM!!! Mr. Tarrou In this math lesson we are

going to be looking at the Counting Principle, Permutations, and Combinations. You might

be getting introduced to these in algebra 2 or precalculus, maybe the beginning of a

probability class, and we do use these extensively through the probability chapter in AP Statistics

another class that I teach. We are going to build the understanding of the counting principle

through building a car choosing from different colors, and types of engines, and transmissions

using a tree diagram. These are really really nice if you want to find the number of combinations,

or since that calculation is pretty simple ultimately, this is really good if you want

to find the probability of a sequence of events. We do this in the probability problems in

AP Statistics. After we have a good understanding of what the counting principle is we are going

to play the lottery. I am going to come up with lottery game that is close to what my

state actually uses. We will try to figure out the probability of us winning the lottery

if we buy one particular ticket, or just one ticket. I don’t know why I said particular.

And through that discussion hopefully help you understand the difference between permutations

and combinations. After I hope that you have an understanding of that, I will go ahead

and give you the formulas for calculating these two and work through two more examples

at the end of the lesson. Now I don’t think that it is a great idea to just have a formula

and have numbers coming at your that you don’t completely understand, but if you do want

to skip ahead to the examples the time stamps will be in the description of this video.

So we want to buy a car. Now, we are at the lot or we are on the internet car building

site and we find that it comes in five different colors with either a straight four cylinder,

a v6, or v8 engine. It will come with either an automatic or manual transmission. I like

manuals transmissions myself if I am driving a sports car. How many unique cars can you

choose from or build on this site? Ok, now you can see, because I already have the notes

written on the board, it is a pretty simple combination. We are going to be able to build

30 unique cars. But if you did not know the counting principle, then you might want to

build a tree diagram to help you understand why there are 30 cars you can build with this

particular setting. Well, if I look at just the automatic transmission. With that auto

we can have a four cylinder, a v6, or a v8. With each of those engine choices we have

white, purple, blue, and orange because that was the colors of chalk I have that show up

the best on this green chalk board. Again you have all those color choices for v6 and

for v8’s. Now if I go to the end of my tree diagram, I have 1,2,3,4, 5, 6, 7, 8, 9, 10,

11, 12, 13, 14, 15 possible cars I can build. That is coming from the three choices I have

for the engine along with the five color choices that I have on out board. That would be one

times three times five, which you can see the beauty of the tree diagram, that there

are ultimately fifteen combinations. Now if I come through here though and say I don’t

want an auto, I want to use a manual transmission well there is another fifteen unique cars

that can build with a manual transmission. So you have two times three times the five

colors give you a total of thirty unique cars you can build for this particular model. Now

of course you can have fancy wheels and plain wheels, and you can have adaptive cruise control.

There are all kinds of choices we have when building cars now, but in this simple example

there are 30 unique cars. Now you don’t want to make a tree diagram every time you want

to find out how many unique in this case vehicles you can build or how many types of ways that

you can build this car. No! You want to just do a simple multiplication like we have at

the bottom of board. So getting my picture in picture out of the way. We are very high

tech here! We have the fundamental counting principle says the number of ways a sequence

of events, key word on a sequence of events… Like you are choosing the color, then you

are choosing the type of transmission, then you are choosing the…umm…the engine. I

think I left that out. …That can occur. This could be for how many different combinations

that does a regular Master combination lock have. You can figure that out. This is found

by multiplying the number of ways or choices each event can occur. You saw that with the

two transmissions, three different engines, different colors. Anything can match with

anything else. Ok, two times three times five. This is sort of a definition for the fundamental

counting principle that is all nice and wordy and hopefully good to understand. You might

see a definition like this in your textbook. If there are m ways to do one thing and n

ways to do another, then there are m times n ways of doing both. Now this wording is

if you only have two events or two things occurring with multiple ways of each one.

But this can just be strung along for whatever. You will see this as we play the lottery.

That is the basic fundamental counting principle. You just multiply along you choices or different

ways that certain events can occur. So if we get that, why don’t we figure out if you

play the lottery, I don’t really think it is a good idea because I know my math, but

hey you can’t win if you don’t play right! So let’s play the lottery. nanananana….

So your state has a lottery that asks you to pick six number one through fifty-three.

How many ways can six unique numbers come up? So you turn the TV on and they are like,

today’s lottery and they got these machines and the balls are flying all over the place,

and they start coming out and they go into that tube one at a time right. They don’t

replace these numbers back into the box, or whatever the machine is called blowing the

air all over the place making the lottery numbers come up. So you have six unique numbers.

One number cannot be played or show up in the game twice. So, as you are looking there

and they have this tube. Usually it is some kind of box like this and this things, not

like that but like that, and those things are bouncing around and there is this big

tube. Just think of that tube and there is six places for those numbers to come up. There

we go. And in the first space every number is inside this machine so any one of those

53 numbers can come up. Then there is only 52 left. Any of those 52 are going to come

up next, and then any of the remaining 51, and then 50, and then 49, and then 48. I am

counting down by one because we are not allowed to repeat these numbers in this particular

example. They have to be unique. So we have one, two, three, four, five, and then finally

our sixth number. Any one of those remaining 48 numbers can come up. Well, using the fundamental

counting principle we know that for each of these events, there are six of them, there

is 53 ways or numbers that can come up for the first event, 52 for the second, 51 for

the third, and so on. We now know that we just simply multiply those together. So right

of the top of my head, 53 times 52 times 51 times whatever comes out

to 16,529,385,600. I almost kept staring at

that number to finally memorize and make you think oooh wow he can do all that in his head.

So there are sixteen, let’s see here this is hundreds, thousands, millions, billions,

there is sixteen billion five hundred twenty-nine million three hundred and eighty-five thousand

six hundred ways that six numbers can be chosen from fifty-three. Awesome. Not really because

like why would I buy a lottery ticket of my odds of winning is one out of sixteen and

half billion. Then you go, wait a minute that seems like a really big number. That can’t

be right. So you go to your, like I live in Florida. You can go to the Florida State lottery

website, whatever that is, and look up the odds of winning these games and yeah indeed,

this is really not the right answer. Now the reason why is, well you have to choose the

right six numbers but they don’t have to be in whatever order they come out of the machine.

Like if these are the winning numbers, that would be weird. If these are the winning lottery

numbers, number 49 could be first and 53 could be next to last. I just need these six numbers,

or any six numbers, and they don’t have to be in order. Well, the fact that the order

of these chosen six does not have to be the same, means that your odds of winning are

a little bit better than what we see here. nanananana… But wait, you don’t again, have

to have the numbers in the correct order, you just need the right numbers. And I did

say the odds of winning were and I did not write out a statement that was appropriate

for the idea of odds. You have to pick one of the 16,529,385,600 different ways that

six numbers can come out of the 53. At least you know, a game might be designed that way.

If you are looking at the number of ways that automobiles are crossing the finish line in

a race, then the order would definitely matter. But in a lottery game it does not. This is

in red because it really is not the odds of winning this particular game if all we have

to do is pick the right six numbers, but not in a particular order. The real odds of winning?

Well, you also need to figure out how many ways these winning numbers can show up. And

this is ultimately going to be the formula for the number of combinations, but I think

it just kind of makes sense if we are developing the idea. So let’s just say for an example

that the winning lottery numbers are 2, 21, 35, 37, 49, and 51. By the way, we may not

know it yet but this is process I did here is the number of permutations, where the order

matters. We have the numbers of 2, 21, 35, 37, 49, and 51. Those are the winning numbers

and let’s say that you picked those winning numbers. Well, when you are watching this

come out on TV, after you know you stop screaming and you play it over again, any one of these

six numbers can come up first. Now we already know that there are a bazillion different

ways that these six numbers can be chosen out of the total of 53 if the order does matter.

But any one of those six winning numbers, if this is the winning series, if this is

what you got on your ticket, well any one of those six can come up first. Any one of

the total of 53 that was in the machine could have come up ok. I am just trying to figure

out this idea of not having to pick the winning numbers in the right sequence. So any of these

six winning numbers could have come up first. And now we cannot repeat any numbers. So now

let’s say that 21came up first, now there is only five more left. Any of those five

could come up next. Then any of those four could come up next. And then three, and then

two, and then when there is only one winning number left to come up, well then no choice

there. There is just that last one. Ok, so now this six times five is thirty. Thirty

times four is 120 and 120 times six, I can do that in my head, is 720. Now that is over

the number, of what I am now going to say, permutations. The way that these six numbers

can come up when the order does matter is going to be 16,529,685,600. Now it just happens

to turn out that both of these numbers, both the numerator and denominator are divisible

by 720. So if we come in here and we reduce the numerator by dividing by 720 and reduce

the denominator by 720, we are going to get an odds of winning which is, where are my

correct set of notes here, one 720 divided by 720 is one. This huge 16 billion and something

divided by 720 comes out to be 22,957,480. So

our odds of winning the lottery game if we

do not have to pick the six winning lottery numbers in the correct order is one out of

twenty-two million nine hundred fifty-seven thousand four hundred eighty combinations

where the order of those six numbers does not make a difference. The number of permutations,

if the sequence did make a difference is quite a bit larger which is again that 16.5 billion

roughly that we had in the first place. So that is the difference between a permutation

and a combination. So let me get this out, add a few more notes, give you proper definitions

and the formulas and finish with two last examples. BAM!!! nanananana… So our formulas.

Permutations. No item can be used more than once, like our lottery numbers. The order

or the arrangement of r items from n matters. That would be nPr is equal to n factorial

over n minus r factorial. The factorial, don’t forget five factorial is five times four times

three times two times one, or maybe I should be counting that the other way around. 5,

4, 3, 2, 1, then you multiply all those numbers together. Don’t forget that zero factorial

by definition is equal to one. The number of Combinations. Again, no item can be used

more than once. The order of arrangements of r items, that would be like again our six

lottery numbers, from n. n would be the size of the population you are pulling from, like

your 53 lottery numbers that were in that thing just flying around waiting for the six

winning lottery numbers to come up. The order of that arrangement does not matter in a combination.

The formula for nCr is n! over (n minus r)! times r! with all those factorials. You will

find these commands in your calculator under the math>probability menu probably if you

have a TI-84 or NSPIRE. Why did I go through that process of talking about playing the

lottery and discussing our way through those scenarios. Because, well we have a lot to

memorize and these formulas don’t really get used that much unless you are in a particular

probability area of mathematics. I just wanted to show you that, yeah it is great that you

know these but we figured those problems out. The number of ways six numbers can come up

out of 53 when the order does matter. We didn’t realize that is what it was. And then we finished

off with that example where oh no, no, no, those six numbers don’t have to show up in

the exact order that you picked them. Then we talked about the number of combinations.

Well let’s see. The number of permutations is n factorial. Now that is going to be, let’s

see. With that lottery and the 53 numbers in the machine all going crazy all over the

place, that was any one of the those 53 numbers could come up. That is n factorial. N, 53

numbers in that machine popping around. Then the denominator is n minus r. Well we were

talking about… Oh and right now I am doing 53nPr6. How many permutations of six numbers

can there be when you are pulling from a total population of 53 of them. I don’t know why

I made the six so much bigger than the 53. So that is 53 minus 6 factorial. Well 53!

is 53 times 52 times 51 times 50, all the way down to one. And 53 minus 6 factorial,

well that is 47! How am I going to do this? 53 factorial is 53*52*51*50, I am going to

run out of room I think, 49*48*47*46…3*2*1. Then the denominator is, I will have to erase

that in a minute, the denominator, this permutation formula says that (53-6)! Well that is 47

factorial. 47 factorial is 47*46*45*…*3*2*1 Counting down until you get to one. All of

a sudden I can’t talk and write at the same time. Well if I go 53 factorial and I count

down by one until I get to one, and if I start at 47 factorial and I count down by one until

I get to one, 47 all the way down to one is going to cancel out with all of these 47’s

down to one. Well 53 times 52 times 51 times 50 times 49 times 48, that is just what we

did in the previous screen developing the number of ways six numbers can be chosen from

a total 53 just using that counting principle, that fundamental counting principle. So this

idea of just having to fill up one, two, three, four, five, six empty spaces where any of

those 53 can come up, and of those 52 can come up, 51, 50, 49, and 48. We sort of did

the permutation formula without even knowing it. So we can kind of figure these out on

our own using a little bit of common sense. Then we talked about, oh wait a minute, these

are all the ways that six numbers kind out of 53 when the order matters. But, then we

said let’s just pretend these are the six winning numbers and talk about how many ways

those six winning numbers can show up. One of those combinations out of these permutations.

That formula is n factorial over n minus r factorial times r factorial. Well that means

that we are going to have, if we are doing 53nCr6, well n factorial. How large is your

population. So we have 53 factorial. This is over n minus r so that is going to be again

53-6 factorial, but now it is times r factorial and we are choosing those six. Now this is

actually upside down from the way I did this in the previous screen. This is just because

of that idea of talking about the odds of winning. It is just natural for me to think

about that being the other way around. But, this is just the number of combinations and

then you are going to pick one of those combinations to get your odds. Well 53 factorial over 47

factorial, we just cancelled that out. It is the same thing as we have up here. It is

53 times 52 times 51 times 50 times 49 times 48. And that 47 all the way down to one cancelled

out with that 47 factorial in the denominator. Here we have the six factorial, which is 6

* 5 * 4 * 3 * 2 * 1. So maybe we have never seen factorials and combinations and we still

have the same looking math, except I had this upside down in the other screen because of

talking about odds of winning. We kind of figured these out on our own without knowing

these formulas. It is nice if you know them and the work goes a little bit faster if you

do. But, you know we can think our way through these problems a little bit right? So now

our last two examples. I am going to verbally speak through them and we can see the work

for these formulas now. So, I am just going to explain how you would think about, and

start them. Then I am going to stop the video, walk off, and I will be done. I will show

you the solutions just before the video ends worked out. You are choosing three people

from a group of twenty to be a part of a committee, for whatever. School safety or something in

your business. How many ways can you select three people? Well think about the scenario.

Is it going to matter what order these three people out of the total of twenty? Is it going

to matter or is it going to not matter? Well if we picking, you have twenty actors and

you are picking three of them to play certain parts it would certainly matter which person

you pick to play which part. They have different lines to memorize. But in this case, if they

are just going to be on a committee and they are all going to kind of just work together

and have no particular job at hand. They are going to have some kind of goal as a committee.

But one person is not going to a boss, and another a secretary, and one is the whatever.

They are just going into a committee. It does not matter how you pick these people. So that

is going to be twenty nCr three. I will show the steps for that solution as we finish up

the video. It is going to be just like this over here. In a race in which there are eighteen

cars, how many ways can the first three cars cross the finish line? I kind of wanted to

do a Nascar race where there is like forty something cars, or endurance racing, or something

like that. Let’s just think of this as a small F1 field because these numbers get big really

fast. So we have 18 cars in the field going around in a loop or on a road course or whatever.

How may different ways can we finish or fill the first three spots, the podium. Well, there

is a huge difference between coming in first, second, or third, right? So in this particular

case the order in which these three cars comes in out of the 18 is a huge difference. This

is definitely a permutation. That is going to be 18 nPr 3. We are just going to work

these formulas out. I hope that seeing how multiplication cancels, and factorials work

is evident from these two problems. I am Mr. Tarrou. Thank you for watching. BAM!!! Go

Do Your Homework.

These vids are very helpful and appreciated. I find textbooks difficult to learn from and I doubt Id be able to progress and explore the wonder of mathematics without people like you. Is there any way I can download/purchase these videos?

Awesome and memorable counting explanation. Well done.

THANKS!!

Hello! Mr.Tarrau you are a real inspiration for me.I am going to sit for GCSE Exam and your videos are really helpful.Thank you very much.I really encourage you to continue further with such helps.

Thank you so much for making these. They help out a ton!

My new #math lesson is now Closed Captioned

i need your brain. lol

do you really reply to every one that comments?

I was actually mocking you, sitting behind a computer, you didn't know that, so…you don't have to thank me…AND I THOUGHT YOU WERE A MALE, "Mrs. T"????

please world teacher ,speak in a way that the one to whom English is a second language to him/her to understand your lessons too

why don't you have a vid about pigeonhole principle? hope u make one in the future. thanks.

Dear Mr Tarrou,

I absolutely love mathematics. Seeing these videos and your mutual love for the subject makes me so happy. You inspire me to continue my dream of teaching and sharing my love of problem-solving with everyone else in a fun and interesting way. Thank you for all that you do. 🙂

Best regards, Hannah Wolf

Do you plan on making videos going over released FRQ questions?

The guidelines they provide does not help with understanding how the problems are done

Can you cover calculus III?

YES! We learned this in my grade 12 data management class a while ago. Wish I'd have found this sooner. This'll be handy before the exam. 😀

Thank you so much! So ready for that quiz tomorrow!

These are actually really cool and engaging, might I add done in very clean and easy to read handwriting. Many props, I really want to see this grow to be something big! I found myself watching videos of yours on topics I wasn't really studying for but picked up rather easily.

Thank you so much for this lesson. I understand the difference between permutations and combinations much better than before. I have a test tomorrow, and I am relying on your videos in addition to my notes.

Can you solve this please?

There are 15 different cars at a car dealership; 6 red cars, 4 white cars, & 5 black cars. What is the number of ways the cars can be parked if none of the white cars can be together?

thanks for this video — God bless ProfBob…

Hi Mr. Tarrou, just wanted to say thanks in general for all the help you've given to me this year! Because of your videos I was able to do well in precalculus and improve in math. You're a great teacher!!

What is the number of ways of expressing 3600 as a product of three ordered positive integers (abc, bca etc. are counted as distinct). For example, the number 12 can be expressed as a product of three ordered positive integers in 18 different ways

You have beautiful handwriting!

Thank you so much for your excellent video. I homeschool our children and 7th grade math has somehow escaped my memory. Today after watching your video not only did we understand the permutations and combinations, but it was fun to learn. Wish I would have had a teacher like you when I was in school. Keep up the wonderful work!

you have beautiful handwriting!!!

cursive?

Actually, the way the final question is worded should give you an answer of 6. If that were a GRE question, the 18 cars would be irrelevant. It is only asking how many ways the first three cars can finish.

how many possible combinations can be made from drawing two cards from a deck consisting of only number cards?

Billions? Your are wrong 1 Billion is 1Thousand of millions, the same 1 million is 1 Thousand of thousands at least in all the world except in your planet. Clearer 1 Billion= 1000 000 000 000 and 1 Trillion= 1000 000 000 000 000 000 and so on. In general great job and excelent calligraphy, It´s just an observation.

THANKSSSSS MANNN!! God bless and more powerrr

can you not write like thomas jefferson please

+ProfRobBob I have question of the same topic but with passwords, for examplePasswords for a new website are created from using either numbers or letters and are not case sensitive and are to be 5 characters long.1.How many total passwords are possible?2.How many passwords are possible if no character can be repeated?3.How many passwords are possible if they must contain at least one vowel?If anybody could explain me or give me a link where somebody explain it, I would appreciate :).

Thanks Prof! 😀

Wow! Thanks for explaining the steps and the reasoning to the formula. I have a better understanding of solving the problems.

thank you !!!

this handwriting can cure cancer

Applause for attempting to explain this. I would bet if 30 average 14 or 15 year old students watched the FCP explanation that is here and then tried to explain it back, only two or three could do it with understanding. Too many new terms, too many examples, haphazardly scribbled on the board. I give you applause for effort and sincerity, but I think you might think about presenting this on Tuesday and testing the class on Wednesday. The results will not be heartening. (One confusing point: The "pie" has numbers in the pieces, yet the numbers have NOTHING to do with the problem. These numbers will distract and confuse some of the students. They don't distract you, the teacher, because you fully understand this.)

thank u soo much f

Thank you! Your writing is so crisp !

I am trying to solve the problem of emails. Let’s say there are three members A, B and C on a team. I see the following email interactions – – A sends a mail to B

– B sends a mail to A

– A sends a mail to C

– C sends a mail to A

– B sends a mail to C

– C sends a mail to B

– A sends a mail to both B and C

– B sends a mail to both A and C

– C sends a mail to both A and BIs this a problem of permutation? If I use the formula for permutation, I get the answer 12 (i.e. 3P2 + 3P3) whereas the above shows only 9 interactions. What am I missing?

Thanks sir. Would learn more if you was our prof.

great

How old are you?

Nananananana}}}

Loved it, Thank you so much. Finally, it got into my head.

Amazing! Thank a lot

So, I have 11 NFL games with 2 teams playing each other with a winner or loser to each game. How many combinations do I have? The Jets are playing the 49ers that week. They are not playing the Gaints therefore the Gaints winning or losing against Jets or 49ers should not be a combination.

I have no common sense when it comes to math – that's why I struggle so much with word problems.

was that mister true? haha

U deserve more subscribes, Clear, Easy, Enjoyable

now I know that lottery is just wasting money when you predict your 6 unique numbers there is only 1% of winning lmao

Oh my god. You saved me! Thanks for explaining this so nicely

Again, thanks endlessly for saving me when it comes to maths

Master locks are permutation locks.

hi i am from india .

where are you from ?

Ur cool 😀

he looks like Indian playback artist sonu nigam

BAMM! You are so dope and cool. I just check out your videos every time they make my work easier ….

Thanks a lot for the help 🙂

You are so organized in teaching!

can anyone help me out? what is the sum of all the numbers formed by digits 1,3,5,7 ( 1 digit, 2 digit…..) without repetition ?

ur handwriting is beautiful

I'm still kinda confused as to why you divide by r! for the Combinations formula.

the permutation and combination thing is really difficult sometimes when the word problems get really elaborate and they dont say specifically "in what kind of ways" do you wish your answer to be given? Like think about your lottery example. The question in itself did ask exactly speaking "in how many ways can 6 unique numbers come up?" Technically speaking this is not the same question as asking "how many combinations of lottery numbers can there be". This is why I think combinatorics can be difficult, it's like you need a lot of "street math" like how lottery works, to be able to do it, like thinking as you go. And the numbers are almost incomprehensibly large, so it's difficult to doublecheck your work also…

amazing

Thanks Mr. Tarrou!

great refresher

It's kind of hard to believe that 10 books on a shelf can be arranged 10! or 3,628,800 ways. 20 books? approx. 2.43 * 10^18 = 2,430,000,000,000,000,000 ways!

Really nice handwriting….

Amazing explanation. Thanks

Am i the only one think that he look like Mr Bean……

#bythewaythankyousir

The handwriting's so beautiful, I'm jealous!!!!

why do teachers not just teach the material and leave all the bullcrap out, it is almost as if they love to hear themselves talk.

You have beautiful handwriting, all my students agree

I never took more than HS math, but being a gambler, I figured all of these out years ago (horses, Lotteries, Casino games, etc), and only bet a buck on the big lottery games when the pot is enormous, and just don't check the ticket for a few days….what you're buying for that dollar is the fantasy of the great things you could do with the money should you win, that is all, and the reason you should never bet more than a buck. You get to go to sleep with the dream fantasy….it isn't an investment for money….just the a cheap way to buy a fantasy. I came up with ((x)(x-1)(x-2)(n-3)(x-4)(x-5))/1*2*3*4*5*6, this if there was 6 numbers drawn and x was the highest number possible. ….if 8 numbers drawn, you would divide by 1*2*3*4*5*6*7*8….it gets fun when you try to figure the least cut game at the casinos. I quit gambling years ago, but figured it was on the craps table, at the casino offering the highest legit odds….a crap table might say "3 times odds" or "5 times odds"…all this means is you can bet 3 times or 5 times the amount of your initial "behind the line" bet at true odds. So if you bet $100 on the "Pass" line, meaning you think the thrower will win, and he doesn't crap out or win on the come out bet, but rather makes a point, (4,5,6,8,9, or 10), instead of betting the 4 say at 9-5, you will get the REAL odds of 2-1, and can bet up to 5 times your initial bet, so in this case, $500. Because you are getting legit odds, you bring down the percentage the house has in its favor. An extremely boring way to bet at a craps table, but the best odds. The only game I know where you can get a true overlay is at the racetrack, but you have to sit out a LOT of races, and be certain you are MUCH better than the other gamblers, because it is a parimutuel pool….the odds are not built in like casino games, but decided by the bets. The track takes out a percentage and the rest is divided among the winners. I love trying to come up with formulas for the different bets, like a superfecta, where you pick the first 4 finishers….if you have 2 to win, 3 for 2nd, 5 for 3rd and 8 for 4th…..but of course, the 2 you chose to win will be in all the others, etc….USUALLY…but you have to know and find a formula that works for ALL possibilities. Someone may just play numbers and not care about picking 2 horses to win, and if 1 wins, they don't use the other in the other spots. I usually wind up charting them out and then backing into a formula. I check my results by using a minimum, and add a few and then a few more to make sure it works for all numbers. When I was a programmer, we tested with garbage, zeros, 1, 10, 100 etc….just to make sure you caught all possiblities as well as kicking out garbage…and always handle zero. That was the old days when you were responsible for everything byte by byte. So many macros now in the higher languages I never worked with, it is just so powerful, but still, programmers should at least understand the logic involved in those macros that someone wrote!!

You're the master of mathematics…….

i salute

Impressed by the way you teach and your writing is awesome

THAT THUMBNAIL

You lost me with all that talking.