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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
2 years 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.

Pi Han Goh - 2 years ago

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