First slide

Binomial theorem for positive integral Index

Question

If sum of the coefficients in the expansion of ${(x + y)}^{n}$ is $2048$ , then the greatest coefficient in the expansion is:

Moderate

A

$^{10} C_{6}$
B

$^{11} C_{6}$
C

$^{10} C_{7}$
D

$^{12} C_{6}$

Solution

Sum of the coefficients in the expansion of ${(x + y)}^{n}$ is $2^{n} .$ Therefore,
$2^{n} = 2048 = 2^{11} \Rightarrow n = 11$ .
As $n$ is odd, the greatest value of $^{n} C_{r}$ is $^{n} C_{k}$ where
$k = \frac{1}{2} (n - 1$ or $k = \frac{1}{2} (n + 1)$ .
Here $n = 11,$ therefore $k = 5$ or $6 .$
Thus, greatest coefficient is $^{11} C_{6}$ .

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