binomial theorem applications

Question

The sum C(n,0) - C(n,1) + C(n,2) - C(n,3) + ... equals

RankGuru · Accepted Answer

1. **Identity Proof**: Consider the expansion of $(1+x)^n$:
   - $(1+x)^n = C(n,0) + C(n,1)x + C(n,2)x^2 + C(n,3)x^3 + \dots$
2. **Substitution**: Let $x = -1$.
   - $(1-1)^n = C(n,0) + C(n,1)(-1) + C(n,2)(-1)^2 + C(n,3)(-1)^3 + \dots$
   - $0^n = C(n,0) - C(n,1) + C(n,2) - C(n,3) + \dots$
3. **Conclusion**: For $n \ge 1$, the alternating sum of binomial coefficients is always zero.

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