Consider the identity: $k(k-1)C(n,k)$=$n(n-1)C(n-2,k-2)$
A- verify the identity for n=10, k=5
B- give an algebraic proof of the identity
C- give a combinatorial proof of the identity.
Can someone help me with this problem. I'm not sure if we plug in exactly the n number and k number into the formula for part A.
For part (a), just plug in and verify. For part (b), I suspect this should be straightforward by just expanding out the combination notation in terms of factorials and cancelling some stuff.
For part (c), consider the problem where out of a group of $n$ people, we need to form a committee of $k$ people with one person being the chair of the committee and another (distinct) person being the vice chair. One way to do this is to first form the committee which can be done in $C(n,k)$ ways, then pick the chair from within the committee ($k$ ways), then pick the vice chair from the remainder of the committee ($k-1$ ways). By the multiplication principle, there are $k(k-1)C(n,k)$ ways to pick everything.
Alternatively, we could first pick the chair from the $n$ people ($n$ ways), pick the vice chair from the remaining people ($n-1$ ways), then form the other $k-2$ members of the committee from the remaining $n-2$ people ($C(n-2,k-2)$ ways). By the multiplication principle, there are $n(n-1)C(n-2,k-2)$ ways to pick everything.
Since the number of ways to form the committee and the chair/vice-chair is the same in both cases, it must be that $k(k-1)C(n,k)=n(n-1)C(n-2,k-2)$.