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Show two to the power of n is the sum of n choose k from k=0 to k=n

$$2^n = \sum_{k=0}^{k=n} \frac{n!}{k!(n-k)!}$$

Note by Jack Han
1 year, 2 months ago

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Hint: What is the binomial expansion of $$(1+x)^n$$? What happens when $$x=1$$?

Alternatively you can prove this via induction. Use the properties of Pascal Identity: $$\dbinom{n-1}{k-1} + \dbinom{n-1}k = \dbinom nk$$.

Or see this: Binomial Theorem - N Choose K. · 1 year, 2 months ago