Combinatorial Proofs

Give a combinatorial proof of the identity $2+2+2 = 3\cdot 2$.

A woman is getting married. She has 15 best friends but can only select 6 of them to be her bridesmaids, one of which needs to be her maid of honor. How many ways can she do this?

1. What if she first selects the 6 bridesmaids, and then selects one of them to be the maid of honor?

2. What if she first selects her maid of honor, and then 5 other bridesmaids?

3. Explain why $6 {15 \choose 6} = 15 {14 \choose 5}$.

Give a combinatorial proof of the identity ${n \choose 2}{n-2 \choose k-2} = {n\choose k}{k \choose 2}$.