Proving 2^n via Induction: 1+n!/1!+n(n-1)!/2!+...”

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The discussion focuses on proving the equation 1 + n/1! + n(n-1)/2! + n(n-1)(n-2)/3! + ... = 2^n through mathematical induction. This expression represents the sum of the binomial coefficients, which corresponds to the power set of a set with n elements, confirming that the number of subsets is 2^n. The binomial expansion of (1 + 1)^n is highlighted as a crucial hint for completing the proof.

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  • Understanding of mathematical induction
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  • Basic concepts of set theory and power sets
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theperthvan
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How can I show that

1+\frac{n}{1!}+\frac{n(n-1)}{2!}+\frac{n(n-1)(n-2)}{3!}+...= 2^{n}

This comes from proving that the power set of a set with n elements is 2^{n}.

I got so far that nCn+nC(n-1)+ ... = what I have above. Now for the induction...
Cheers,
 
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Hint: What's the binomial expansion of (1 + 1)^n?
 
of course, thanks
 

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