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Binomial Coefficients Identity

  • Thread starter murmillo
  • Start date
  • #1
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Homework Statement


Prove that for an integer n greater than or equal to 2,

nC1 - 2nC2 + 3nC3 - + ... = 0. (nCm means n choose m)

Also,
2x1 nC2 + 3x2 nC3 + 4x3 nC4 +... = n(n-1)2^(n-2)

Homework Equations



(1+t)^a = 1 + aC1(t) + aC2(t^2) + ...

The Attempt at a Solution


I don't know if these identities will help, but I've found
nC0 - nC1 + nC2 - nC3 + - ... = 0
and
nC0 + nC1 + nC2 +... = 2^n

I tried writing out the given expression in terms of factorials and got
1/0! n - 1/1! n(n-1) + 1/2! n(n-1)(n-2) - 1/3! n(n-1)(n-2)(n-3) + - ...,
but I don't think this is going anywhere.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
The binomial theorem is your friend. (1+x)^n=nC0+nC1*x+nC2*x^2+...+nCn*x^n, right? Think about what you might want to put x equal to, and think about taking derivatives of (1+x)^n.
 

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