In the expansion of $(a + b)^{n}$ , the coefficients of 15th and 11th terms are equal, Find the number of terms in the expansion.

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Solution

The correct option is D 24
Given, $(a + b)^{n}$
General term
$T_{r + 1} =^{n} C_{r} (x)^{n - r} (y)^{r}$
$T_{r + 1} =^{n} C_{r} a^{n - r} b^{r}$

given, the coefficient of $15$ th and $11$ th terms are equal
$T_{15} = T_{14 + 1} =^{n} C_{14} a^{n - 14} b^{14}$
$T_{11} = T_{10 + 1} =^{n} C_{10} a^{n - 10} b^{10}$
$^{n} C_{14} =^{n} C_{10}$
$\frac{n!}{(n - 14)! 14!} = \frac{n!}{10! (n - 10)!}$
$10! (n - 10)! = 14! (n - 14)!$
$10! (n - 10)! = 14 \times 13 \times 12 \times 11 \times 10! (n - 14)!$
$(n - 10) (n - 11) (n - 12) (n - 13) (n - 14)! = 14 \times 13 \times 12 \times 11 (n - 14)!$
$(n - 10) (n - 11) (n - 12) (n - 13) = 14 \times 13 \times 12 \times 11$
Using options:
$n = 24$
So option D is correct.

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In the expansion of (a+b)n, the coefficients of 15th and 11th terms are equal, Find the number of terms in the expansion.

In the expansion of $(a + b)^{n}$ , the coefficients of 15th and 11th terms are equal, Find the number of terms in the expansion.