## 15 Aug 2011

### Prove this identity

A note to future me:

Today you were trying to do probability questions with your pathetically sad probability skills and you found that
$\sum_{i=0}^{n-1}\frac{m}{t-i}\frac{\binom{t-m}{i}}{\binom{t}{i}}=1-\frac{\binom{t-m}{n}}{\binom{t}{n}}.$
Dare you to prove it.

From,
Your arrogant younger self.

Here's a hint:
$\frac{\binom{t-m}{i}}{\binom{t}{i}}=\frac{\binom{t-i}{m}}{\binom{t}{m}}.$
Why? Can you think of a combinatorial interpretation for this? Proving it through algebra is trivial. Thinking of an interpretation may lead to some actual learning.

Bonus:
From the identity above, you also managed to find that it implies: $\sum _{i=0}^n \binom{t-i}{m}=\binom{t+1}{m+1}-\binom{t-n}{m+1}.$ Prove this identity.