For this week I was asked to prove the following by mathematical induction:
$\sum_{k=0}^m {{m}\choose{k}}y^k \leq (m+1)!$ with $0 \leq y \leq 1$.
Now for my P(M+1) part, I have tried to merely replace m by it, such that I obtain: $\sum_{k=0}^{M+1} {{M+1}\choose{k}}y^k$ and then using the fact that ${{M+1}\choose{k}}={{M}\choose{k}}+{{M}\choose{k-1}}$ has not yielded the desired result. Is there anyone who could help me a step in the right direction? I just have no clue what to do, to be honest. I am pretty bad with thinking in terms of binomial coefficients (and especially whenever you do a summation from 0 to M).
Hopefully someone is willing and able to help me out. Thank you in advance.