Question

If the number of subsets with 4 elements of a set A is equal to the number of subsets with 5 elements of the set, thenthe number of subsets with 3 elements of this set is

Answer 1

The formula for the number of subsets with \(m\) elements from a set of \(n\) elements is given by\[\binom{n}{m}=\frac{n!}{m!(n-m)!}\]So you're given the information that there are the same number of 4-element and 5-element subsets. This tells you that\[\binom{n}{4}=\binom{n}{5}\]and you want to find \[\binom{n}{3}.\]

Answer 2

So let's try to solve for \(n\). We know that\[\binom{n}{4}=\binom{n}{5}\]or that\[\frac{n!}{4!(n-4)!}=\frac{n!}{5!(n-5)!}.\]Multiplying by \(5!\), this results in the equation\[\frac{5n!}{(n-4)!}=\frac{n!}{(n-5)!}.\]Now we can cancel the \(n!\) from both sides, and multiply by \((n-4)!\) to get\[5=\frac{(n-4)!}{(n-5)!}=n-4\implies n=9\]

Answer 3

Thus, \(n=9\), and so we only have to calculate\[\binom93=\frac{9!}{3!6!}=84\]