Greatest Coefficient

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Among all the coefficients appearing in the binomial expansion of (1 + x)^{15}, the greatest coefficient is attained at the terms whose binomial coefficient equals which value?

RankGuru · Accepted Answer

In the expansion of (1 + x)^n, the coefficient of the term is simply C(n, r), and the greatest binomial coefficient occurs at the middle of the row, a result rooted in the unimodality of Pascal's triangle that JEE Advanced frequently exploits. For odd n = 15, there are two equal greatest coefficients, namely C(15, 7) and C(15, 8), since the row is symmetric and peaks at the two central positions. Computing C(15, 7) = 15! / (7! 8!) = 6435, which equals C(15, 8) by symmetry. Hence the greatest coefficient is 6435. Option 5005 = C(15, 6) is one step away from the centre and therefore smaller. Option 3003 = C(15, 5) is even further from the middle. Option 1365 = C(15, 4) is smaller still. The sequence C(15, r) strictly increases up to r = 7 and then decreases, confirming the maximum at the centre. Plausibility check: the sum of all coefficients is 2^{15} = 32768, and the largest single coefficient 6435 is a sizeable but sub-half fraction of this, which is consistent with a broad, bell-shaped distribution of coefficients.

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