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JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Binomial Theorem

The mean of the coefficients of

x, x^{2}, \dots, x^{7}

in the binomial expansion of

(2 + x)^{9}

is

Answer: 2736

Solution

The coefficient of

x^{r}

in

(2 + x)^{9}

is

(r 9) 2^{9 - r}

. For

r = 1, 2, \dots, 7

, the coefficients are

(1 9) 2^{8}, (2 9) 2^{7}, \dots, (7 9) 2^{2}

. Their sum is

(2 + 1)^{9} - (0 9) 2^{9} - (8 9) 2^{1} - (9 9) = 3^{9} - 512 - 18 - 1 = 19683 - 531 = 19152

. Mean

= \frac{19152}{7} = 2736

.

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Integer answer

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