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

$^{31} C_{5}$
B

$^{3} C_{5}$
C

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

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

Solution

$(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30} = (1 + x)^{21} [\frac{(1 + x)^{10} - 1}{(1 + x) - 1}] = \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}] \Rightarrow Coefficient of x^{5} in the given expression = Coefficient of x^{5} in \{\frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]\} = Coefficient of x^{6} in [(1 + x)^{31} - (1 + x)^{21}] =^{31} C_{6} -^{21} C_{6}$

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