The coefficient of $x^{r} (0 \leq r \leq (n - 1))$ in the expansion of ${(x + 3)}^{n - 1} + {(x + 3)}^{n - 2} (x + 2) + {(x + 3)}^{n - 3} {(x + 2)}^{2} + \dots + {(x + 2)}^{n - 1}$ is

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$^{n} C_{r} (3^{r} - 2^{n})$

$^{n} C_{r} (3^{n - r} - 2^{n - r})$

$^{n} C_{r} (3^{r} + 2^{n - r})$

$^{n} C_{r} (3^{r} - 2^{n - r})$

answer is B.

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Detailed Solution

${(x + 3)}^{n - 1} + {(x + 3)}^{n - 2} (x + 2) + {(x + 3)}^{n - 3} {(x + 2)}^{2} + \dots + {(x + 2)}^{n - 1}$

we know that

$a^{n} - b^{n} = (a - b) (a^{n - 1} + a^{n - 2} b + \dots + b^{n - 1})$

Given expression

$\frac{{(x + 3)}^{n} - {(x + 2)}^{n}}{(x + 3) - (x + 2)} = \frac{{(x + 3)}^{n} - {(x + 2)}^{n}}{1}$

Coefficient of $x^{r}$ in ${(x + 3)}^{n} - {(x + 2)}^{n}$

$\begin{array}{l} ^{n} C_{n - r} 3^{n - r} -^{n} C_{n - r} 2^{n - r} \\ =^{n} C_{r} (3^{n - r} - 2^{n - r}) \end{array}$

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