# Counting number of subsets with restriction

Let $x\geq 66$ be an integer and consider set $S = \{1,2,3,4....,x\}$

(1) $k$ is an integer with $66\leq k\leq x$. How many $66$-element subsets of $S$ are there whose largest element is equal to $k$?

(2) Use the answer in first part to prove -

$$\sum {k-1\choose 65}={x\choose 66}$$

I think the answer of part 1 is $k$ chooses $66$ because that way $k$ will be in all subsets. Since $k \leq 66$, $k$ will be the largest in each subset. Is there anything wrong with my logic? For part 2, I'm not sure how to go about proving that equation. Can anyone guide me to the right direction? Thanks.

• Please check if my edit has changed your question. I'm not exactly sure about part 2. Jan 29 '14 at 7:20
• Yes, that's my question. Thank you! Jan 29 '14 at 8:11

To be concrete, let's take the example of $k = 100$.
If you want to choose a subset with $66$ elements whose largest element is $100$, what are you allowed to choose freely?
• Yes. How many numbers are there less than $k$? Jan 29 '14 at 8:12