First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

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

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

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

$n_{c_{r}} (3^{r} - 2^{n - r})$

Solution

We have
$(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots + (x + 2)^{n - 1}$
$= \frac{(x + 3)^{n} - (x + 2)^{n}}{(x + 3) - (x + 2)} = (x + 3)^{n} - (x + 2)^{n}$ $(∵ \frac{x^{n} - a^{n}}{x - a} = x^{n - 1} + x^{n - 2} a^{1} + x^{n - 3} a^{2} + \dots + a^{n - 1})$
Therefore, coefficient of x^r in the given expression is equal to coefficient of x^r in [(x + 3)ⁿ - (x + 2)ⁿ], which is given by
$^{n} C_{r} 3^{n - r} -^{n} C_{r} 2^{n - r} =^{n} C_{r} (3^{n - r} - 2^{n - r})$

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