Verified Solution • 2025–26 Session moderate

QJEEM25Ma0828: JEE Main 2025 Mathematics - Class 11 JEEM25Ma0828 Solution: Suppose and are the coefficients of and terms respectively in…

Suppose AA and BB are the coefficients of 30th 30^{\text {th }} and 12th 12^{\text {th }} terms respectively in the binomial expansion of (1+x)2n−1(1+x)^{2 \mathrm{n}-1} . If 2 A=5 B2 \mathrm{~A}=5 \mathrm{~B} , then n is equal to:

Multiple Choice Options

  • A 20
  • B 19
  • C 22
  • D 21

Step-by-Step Verified Solution

amp;A=2n−1C29B=2n−1C11amp;22n−1C29=52n−1C11amp;2(2n−1)!29!(2n−30)!=5(2n−1)!(2n−12)!11!amp;129…12⋅5=1(2n−12)(2n−13)…(2n−29)2amp;130⋅29…12=1(2n−12)(2n−13)…(2n−29)12amp;2n−12=30amp;n=21\begin{aligned}& A={ }^{2 n-1} C_{29} \quad B={ }^{2 n-1} C_{11} \\& 2{ }^{2 n-1} C_{29}=5{ }^{2 n-1} C_{11} \\& 2 \frac{(2 n-1)!}{29!(2 n-30)!}=5 \frac{(2 n-1)!}{(2 n-12)!11!} \\& \frac{1}{29 \ldots 12 \cdot 5}=\frac{1}{(2 n-12)(2 n-13) \ldots(2 n-29) 2} \\& \frac{1}{30 \cdot 29 \ldots 12}=\frac{1}{(2 n-12)(2 n-13) \ldots(2 n-29) 12} \\& 2 n-12=30 \\& n=21\end{aligned}

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