Greatest Coefficient
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?
Select the correct option:
Solution
6435
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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About This Question
- Subject
- mathematics
- Chapter
- binomial theorem and its simple applications
- Topic
- greatest coefficient
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
6435
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.
This medium difficulty mathematics question is from the chapter binomial theorem and its simple applications, covering the topic of greatest coefficient. It appeared in the 2025 exam.
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