psi20 Posted March 20, 2005 Share Posted March 20, 2005 I'm trying to prove that, hopefully this comes out right, [math]\sum_{r=0}^{n}{(-1)}^{r}{_n}C{_r}{(n+1)}^n = n![/math] Can anyone help? The pattern is from 1 4 9 3 5 2 1 8 27 64 7 19 37 12 18 6 1 2 1 etc. Link to comment Share on other sites More sharing options...
uncool Posted March 20, 2005 Share Posted March 20, 2005 I'm not sure about the sum part, but the others come from taking differences. Here iswhy it works: ((x+1)^n-x^n) = nx^(n-1)+n*(n-1)*x^(n-2)/2 + ... So the coefficient of the first term will be n. You then subtract again, because you eventually want to get a constant number times the first term without the x value. This leaves you with n*(n-1). Then you subtract again, leaving you with n*(n-1)*(n-2) . . . Then you subtract again, leaving you with n*(n-1)*(n-2)*....*3*2*1 = n! All the other coefficients cancel out. Hope this helps. -Uncool- Link to comment Share on other sites More sharing options...
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