Some Identities explained

sum(C(n,k)^2, k=0..1) = C(2n/n). The left hand side is interpreted as n men and n women and we need to find a group of n people. Thus left side is when you choose k men then you need to choose (n-k) women and since C(n,k) = C(n,n-k). Therefore C(n,k)*C(n,n-k) = C(n,k)^2. The right side is obvious choosing n people from 2n people.

k*C(n,k) = n*C(n-1,k-1). The left side can be interpreted as choosing k people and then choosing the leader from those k people. The right hand side is first choosing the leader from n people which can be done in n ways and then choosing the remainning k-1 people which can be done in C(n-1,k-1) and hence together can be done in n*C(n-1,k-1)