Solution: 2016 Fall Midterm - 2

Author: Michiel Smid

Question

Let $k$ and $n$ be integers with $2 \leq k \leq n$ and consider the set $S = \{1,2,\dots,n\}$. What is the number of $k$-element subsets of $S$ that do not contain 1 and do not contain 2?

(a)

${n - 1 \choose k}$

(b)

${n - 1 \choose k - 1}$

(c)

${n - 2 \choose k - 2}$

(d)

${n - 2 \choose k}$

Solution

Well, we take away 1 and 2 from the bag that we get to choose from

Thus, we need to choose $k$ elements from $n-2$ elements.

There are $\binom{n-2}{k}$ ways to do this.