Solution: 2019 Fall Midterm - 2

Author: Michiel Smid

Question

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

(a)

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

(b)

${n - 2 \choose k}$

(c)

${n - 1 \choose k}$

(d)

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

Solution

Um, let’s just take 1 and 2 out of the set

So this means we have $n-2$ elements left in the set.

We still need to choose $k$ elements from the $n-2$ elements, so the answer is $\binom{n-2}{k}$.