Sum of Binomial Coefficients

#Math #Binomial #Algebra

In this video we solve a problem that involves binomial coefficients.

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I share Maths problems and Maths topics from well-known contests, exams and also from viewers around the world. Apart from sharing solutions to these problems, I also share my intuitions and first thoughts when I tried to solve these problems.

[SimsHacks]

Oh, one of rare problems I can solve 🤣

[Artisharma]

(1+x)^n = summation of nCr ×1^r × x^n-r, r belongs from 0 to n. So just put x=1 and boom👊 we get 2^20

[sjaubain]

The problem is similar to finding all possible combinations of items in a "bag" of n items. Another way to solve it would be to imagine a binary tree with height n and for each step starting from root, go left if we pick the kth item or right if we don't. There is finally 2^n leafs representing all the possibilities :)

[jarikosonen]

Maybe from symmetry it can be seen than the requested sum would be 1/2 of the sum of all the binomial coefficients.
It looks correct. Maybe this is better for even exponents.
Also induction can be used to show each row sum is 2^n.

[math]

was able to solve this, thanks for sharing

[stkhan]

.hi letsthinkcritically ..make a pdf available..beautiful

[jackbridge]

What a long explanation. Is it not easier to just consider a binary number of 20 digits. Each term is just the number of ones. So you get all possible numbers which is 2^20.