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Application of Stirling's Formula

An often used application of Stirling's approximation is an asymptotic formula for the binomial coefficient. One can prove that for k = o(n exp3/4),

(n "choose" k) ~ c(ne/k)^(k)

for some appropriate constant c. Can you find the c? Can you say why this only works when k is much smaller than n exp3/4?

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Stirling's approximation is applied. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.