Solution: 2019 Winter Midterm - 7

Author: Michiel Smid

Question

Consider 17-element subsets of the set $\{1,2,3,\dots,45\}$.
How many such subsets have the property that the largest element in the subset is equal to 30?

(a)

${{29}\choose{16}}$

(b)

${30 \choose 17}$

(c)

${29 \choose 17}$

(d)

${30 \choose 16}$

Solution

We can treat the largest element as a fixed element.

We need to choose 16 other elements from the remaining 29 elements that are smaller than 30: $ \binom{29}{16} $