First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{r}$ in the expansion of $\begin{array}{l} S = (x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1} \end{array}$ is

Moderate

A

$3^{n - r} - 2^{n - r}$
B

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

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

none of these

Solution

We can write $S = \frac{(x + 3)^{n - 1} [1 - {(\frac{x + 2}{x + 3})}^{n}]}{1 - \frac{x + 2}{x + 3}} = (3 + x)^{n} - (2 + x)^{n}$
$∴$ Coefficient of $x^{r} =^{n} C_{r} (3^{n - r} - 2^{n - r})$

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