I have to show the following: C(n,0) + C(n+1,1) + ...+ C(n+r,r)= C(n+r+1,r) The
ID: 1889805 • Letter: I
Question
I have to show the following:
C(n,0) + C(n+1,1) + ...+ C(n+r,r)= C(n+r+1,r)
The original problem is written the other way combinations are written as but the equation didn't show up correctly so I used this alternatively.
Also, anyone have any advice for solving permuations and combinations problems? For some reason I cannot seem to grasp these concepts fully. I understand the broad scheme but when I am faced with problems, I freeze because I feel like I don't know what to do. Anything that could make it easier to understand these problems fully would be greatly appreciated especially since my whole class is applied combinatorics. Thanks.
Explanation / Answer
see nCr indicates the no. of possibilities of choosing r objects from n different objects. now this prblm : nC0 + n+1C1 + ......+n+rCr denotes of choosing 0 objects from n objects then choosing 1 object from n+1 objects then choosing 2 objects from n+2 objects and so on.. and adding those possibilities now see this C(n+ r + 1, r) = C(n + r, n) + C(n+ r - 1, r - 1) + ... + C(n, 0). For r = 0, C(n + 1, 0) = 1 = C(n, 0), the only term on the right.Let r = k + 1: C( n+k + 2, k + 1= C(n+k + 1, k + 1) + C(n+k+ 1, k) - Identity (Pascal's Triangle) = C(n+k+ 1, k + 1) + C(n+k, k) + C(n+k - 1, k - 1) + ... + C(n, 0).