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(C) The number of selections is the coefficient of $x^8$ in the expansion of $(1 + x + x^2 + \dots + x^8)(1 + x + x^2 + \dots + x^8)(1 + x)^8$.This is

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$10 \cdot 2^7$

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(C) The number of selections is the coefficient of $x^8$ in the expansion of $(1 + x + x^2 + \dots + x^8)(1 + x + x^2 + \dots + x^8)(1 + x)^8$.This is the coefficient of $x^8$ in $\left(\frac{1 - x^9}{1 - x}\right)^2 (1 + x)^8$.Since we are looking for the coefficient of $x^8$,we can ignore the $(1 - x^9)^2$ term as it contributes only $1$ to the coefficient of $x^8$.Thus,we need the coefficient of $x^8$ in $(1 - x)^{-2} (1 + x)^8$.$(1 - x)^{-2} = 1 + 2x + 3x^2 + \dots + 9x^8 + \dots$$(1 + x)^8 = \sum_{r=0}^{8} {}^8C_r x^r = {}^8C_0 + {}^8C_1 x + {}^8C_2 x^2 + \dots + {}^8C_8 x^8$.The coefficient of $x^8$ is $\sum_{r=0}^{8} (r+1) {}^8C_r = {}^8C_0 + 2{}^8C_1 + 3{}^8C_2 + \dots + 9{}^8C_8$.Consider $f(x) = \sum_{r=0}^{8} {}^8C_r x^{r+1} = x(1 + x)^8$.Differentiating with respect to $x$: $f'(x) = \sum_{r=0}^{8} (r+1) {}^8C_r x^r = (1 + x)^8 + 8x(1 + x)^7$.Setting $x = 1$,we get $\sum_{r=0}^{8} (r+1) {}^8C_r = (1 + 1)^8 + 8(1)(1 + 1)^7 = 2^8 + 8 \cdot 2^7 = 2^7(2 + 8) = 10 \cdot 2^7$.

The number of ways of selecting $8$ letters from $24$ letters,of which $8$ are $a$,$8$ are $b$,and the remaining $8$ are distinct,is given by:

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