The Empire is attacking a Rebel base that is stocked with [tex]n[/tex] X-wings and [tex]n[/tex] Y-wings. The Rebels need to build a fleet consisting of [tex]n[/tex] ships (with at least 1 X-wing), to be led by 1 pilot in an X-wing.
There are [tex]\binom n1=n[/tex] ways of picking the leader, and [tex]\binom{2n-1}{n-1}[/tex] ways of building the rest of the fleet, so there's a total of
[tex]n\dbinom{2n-1}{n-1}[/tex]
ways of building such a fleet.
In the other direction, suppose we build a fleet comprising of [tex]k[/tex] X-wings and [tex]n-k[/tex] Y-wings. We have [tex]\binom nk[/tex] ways of picking X-wings and [tex]\binom n{n-k}[/tex] ways of picking Y-wings. Also from the [tex]k[/tex] X-wings we pick 1 to be the leader, which we can do in [tex]\binom k1=k[/tex] ways. So there are
[tex]k\dbinom nk\dbinom n{n-k}[/tex]
ways of building such a fleet. But since
[tex]\dbinom nk=\dbinom n{n-k}[/tex], we have
[tex]k\dbinom nk^2[/tex]
ways of building the fleet with these specifications. Sum over all possible values of [tex]k[/tex],
[tex]\displaystyle\sum_{k=1}^nk\binom nk^2[/tex]
