To the person answering this, can I get an explanation for each step, because I want to learn how to solve this problem and future problems that are like this one. Thanks!
The problem is I don't understand his solution.
But I still don't understand how N choose K-1 = \(n!/(k-1)!(n-k+1)!\).
Why is it +1 Instead of -1?
Well, have you tried a numerical example? Try this:
4 choose 2 =6. Now, make it like this: 4 choose (3 - 1) . Try substituting 4 choose (3-1) for n choose (k-1) and see how to get the same result as 4 choose 2 = 6. When he shows: n - k + 1, he is actually doing this: (n - (k - 1)). First you would do the innermost brackets:(n - k + 1), because - * - = +. So that 4 choose (3-1) gives you exactly the same result as 4 choose 2 = 6 . Do you understand it now?
So I have one last question, why is this true, \((k-1)!=k!/k\).