I understand the combinatorial proof and the common sense behind why it works but lately I am trying to play around with proofs since I am still new to them. So I understand part of this: http://www.google.com/imgres?imgurl...QIUvOZD4K6yQHDp4Bw&ved=0CGAQ9QEwBjgK&dur=1607 Where I am getting confused is the step where we combine terms. The denominator of the first term is (n-k+1)k!(n-k)! , and the denominator of the second term is k(k-1)!(n-k+1)! Then on the next line the denominator is k!(n-k+1)!. How is that a common denominator for those two terms? I see there is a (n-k+1)! in both terms.....but in the second term there is a k(k-1)! and in the first term there is a k!(n-k)!....how do those two terms somehow both reduce to k!?