For C(n, m), what values of n and m make C the largest?

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The values of n and m that maximize C(n, m) depend on the interpretation of C(n, m). If C(n, m) refers to the binomial coefficient, there is no largest value as increasing n indefinitely yields larger coefficients. For a fixed n, the maximum value of C(n, m) occurs at m equal to n/2 when n is even, or at m equal to (n-1)/2 and (n+1)/2 when n is odd. Observing Pascal's triangle reveals the pattern for determining the optimal m for a given n. Understanding these relationships is crucial for maximizing the binomial coefficient.
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for C(n, m), what values of n and m make C the largest?
 
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Well, that depends a lot on what C(n,m) means!

If it is the binomial coefficient, \frac{n!}{m!(n-m)!}, then the answer depends on precisely what you are asking. If I take your question literally: that n and m can be any positive integers (m<= n) there is no answer: taking n larger and larger gives larger and larger values for C(n,m). There is no largest value.

If you mean "for a specific n, what value of m makes the binomial coefficient C(n,m) largest", write a few rows of Pascal's triangle and the pattern should become obvious. A precise answer then will depend on whether n is even or odd.
 
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