The mean of the coefficients of x , x 2 , … , x 7 in the binomial expansion of ( 2 + x ) 9 is __________. [2023]

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Solution

Q.
The mean of the coefficients of $x, x^{2}, \dots, x^{7}$ in the binomial expansion of ${(2 + x)}^{9}$ is __________. [2023]

Ans.
(2736)

${(2 + x)}^{9} = {}^{9}C_{0} 2^{9} + {}^{9}C_{1} 2^{8} x^{1} + {}^{9}C_{2} 2^{7} x^{2} + {}^{9}C_{3} 2^{6} x^{3} + {}^{9}C_{4} 2^{5} x^{4} + {}^{9}C_{5} 2^{4} x^{5} + {}^{9}C_{6} 2^{3} x^{6} + {}^{9}C_{7} 2^{2} x^{7} + {}^{9}C_{8} 2 x^{8} + {}^{9}C_{9} x^{9}$

$Coefficient of x = {}^{9}C_{1} 2^{8}$

$Coefficient of x^{2} = {}^{9}C_{2} 2^{7}, \dots, coefficient of x^{7} = {}^{9}C_{7} 2^{2}$

$Mean = \frac{{}^{9}C_{1} \cdot 2^{8} + {}^{9}C_{2} \cdot 2^{7} + \dots + {}^{9}C_{7} \cdot 2^{2}}{7}$

$= \frac{{(1 + 2)}^{9} - {}^{9}C_{0} \cdot 2^{9} - {}^{9}C_{8} \cdot 2^{1} - {}^{9}C_{9}}{7}$

$= \frac{3^{9} - 2^{9} - 18 - 1}{7} = \frac{19152}{7} = 2736$

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