Combinations Formula: Counting the number of ways to choose r items from n items.

We have previously looked at the permutation formula to PICK r items from n items, when we care about the order they come out. Now we look at how to CHOOSE r items from n items, where we don't care about the order they come out. We derive the combination example formula and do an example

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[ArleynH]

This is legit one of the best explanations of this particular topic I’ve ever seen. Thank you for your videos!

[augustobatista]

Highschool, book, teacher none of those were clear as this props to being one of the best math channels, I am a computer engineering major and every time that I get stuck I already know where to find an answer. Thank you!

[andrewharrison]

Not sure why this video has so few upvotes.
Yes, I knew about combinations but it is still good to see your explanation without frills. It deserved 6 minutes, you gave it six minutes - job done.
Thanks

[badasskiker]

You saved my life dude tysm. Please never stop making these videos.

[continnum_radhe-radhe]

I try to understand this problem from high school ...but i was not understand .
Finally i understand this concept..
Thank you very much sir ❤️🔥❤️

[tiffanykawamura]

Only video that actually helped!! Thank you so much. Don’t stop teaching and making vids :)

[ramyhuber]

OK am more than 3/4 through this course, really enjoying it, this titled course didn't exist way back when I had a math minor at university. You're an excellent teacher, thank you!

[BoZhaoengineering]

The presence of all wanted elements from sample space suddenly makes the algebraic fraction removal holds strong and legitimately. Thank you for your clear interpretation.

[continnum_radhe-radhe]

I think memorise first and then understand this helps to know what we are understanding...

[mauriciorey]

Thanks Trefor for doing these videos, you are an inspiring professor.

[cschandragiri]

Very well explained between choose and pick!

[andrewharrison]

The number of combinations is obviously an integer.
If we choose n objects from n+m items then we have (n+m)!/(n!.m!) possibilities so this must be an integer.
If we cancel the last m terms of (n+m)! with the m! in the denominator and look at what is left:
The product of n consecutive integers is divisible by n!
So if we multiply the 9 numbers from 11 to 19 then it ought to be divisible by 9! even though 11, 13, 17 and 19 don't help at all.

Well, I am convinced that's true but feel that it ought to be possible to prove it directly. This lead me in the direction of modular arithmetic and prime factorisation - which feels weird when we started with combinations.

If x/n gives remainder r then (x+1)/n gives remainder r+1 this is mod n so if r+1 is n we get 0. So we can see that over n terms the remainder when dividing by n has values from 0 to n-1 in order but not usually starting at 0.

The remainders for dividing by numers less than n have the same pattern but repeating e.g. 11 to 19 divided by 6 is 5, 0, 1, 2, 3, 4, 5, 0, 1

So each number from 1 to n divides into one or more of the n consecutive numbers but all of them at the same time? Consider each prime factor of n! in my example 3 is in 9 (twice), 6 and 3. In the n consecutive terms the largest power of the prime p (e.g. 9) will divide one of the terms (e.g. 18) and the other numbers containing p and offset in both directions will be enough other multiples of p to match the powers of p in n!.

So for 3 in the example the 9 lines up with 18, the 6 lines up with 15 (so the 3 cancels but not the 2) and the 3 lines up with 12.
It's fun to line up the 2s as well starting with 8 lining up with 16 and going both ways:
18, 16, 14, 12
2, 8, 2*, 4
* the 2 from the 6

[georgesnganou]

This was a great explanation, but one thing I have yet to see or hear is to explain why dividing???
There are 4 basic operations. Adding and Multiplying doesn't make sense, but I can see a debate between subtracting and dividing.

[rishitshah]

Sir it was really great explanation
I really like it
i always get one doubt which one is 'n' and which one is 'r'
for example in the 3 digit lock which one is 'n' and which one is 'r'
please help me sir
Thank you
regards
Rishit