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Rainbow Cupcake

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**Summary::**prove that (n 0) + (n 1) + (n 2) + ... + (n n) = 2^n is true using mathematical induction.

note that (n n) is a falling factorial

Hello! I have trouble dealing with this problem:

**Mod note: Thread moved from math technical section, so is missing the homework template.**

Prove that

**(n 0) + (n 1) + (n 2) + ... + (n n) = 2^n**is true using mathematical induction.

note that

**(n n) is a falling factorial.**I have made some progress with this problem but I got eventually got stuck.

progress:

let n=1 (please see attached photo)

Therefore it is true for n=1

Then we assume that (k 0) + (k 1) + ... + (k k) = 2^k is true. This could also be written as

(k 0) + (k 1) + ... + (k k-1) + (k k) = 2^k.

Prove that (k+1 a) = (k a-1) + (k a) (pls see attached photo for progress)

After this step, i got stuck.

Thank you!

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