SOLUTION: Can someone, please, explain why this equation is true? {{{sum( (-1)^k * (matrix (2, 1, n, k)), k=0, n) = 0}}} I intuitively understand why it works for odd n (as combination

Algebra ->  Permutations -> SOLUTION: Can someone, please, explain why this equation is true? {{{sum( (-1)^k * (matrix (2, 1, n, k)), k=0, n) = 0}}} I intuitively understand why it works for odd n (as combination      Log On


   

Question 56961: Can someone, please, explain why this equation is true?
sum%28+%28-1%29%5Ek+%2A+%28matrix+%282%2C+1%2C+n%2C+k%29%29%2C+k=0%2C+n%29+=+0
I intuitively understand why it works for odd n (as combinations with k elements are as many as combinations with n-k elements and they just cancel each other out in the end). But this doesn't explain the equation for even n.
Thanks!
Found 2 solutions by stanbon, fanks:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Sure it does 8C6=8C(8-6).
The evens "add" out also.
Try it for a small n like n=8 and
see what it looks like.
Cheers.
Stan H.

Answer by fanks(6) About Me  (Show Source):
You can put this solution on YOUR website!
Thanks for the reply, Stan!
However, they don't cancel each out, as, for example, both %28matrix+%282%2C+1%2C+8%2C+6%29%29 and %28matrix+%282%2C+1%2C+8%2C+8-6%29%29 would have a plus sign in that sum (%28-1%29%5E6 and %28-1%29%5E2).