First slide

Binomial theorem for positive integral Index

Question

The coefficient of x⁵ in the expansion of $(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$ is

Moderate

A

$^{51} C_{5}$
B

$^{9} C_{5}$
C

$^{31} {C_{6}}^{- 21} C_{6}$
D

$^{30} C_{5} +^{20} C_{5}$

Solution

$(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$
$\begin{aligned} = (1 + x)^{21} [1 + (1 + x)^{1} + \dots + (1 + x)^{9}] \\ = (1 + x)^{21} [\frac{(1 + x)^{10} - 1}{(1 + x) - 1}] \\ = \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}] \end{aligned}$
$∴$ Coefficient of f in the given expression
$\begin{array}{l} = Coefficient of x^{5} in \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}] \\ = Coefficient of x^{6} in [(1 + x)^{31} - (1 + x)^{21}] \\ =^{31} C_{6} -^{21} C_{6} \end{array}$

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