First slide

Binomial theorem for positive integral Index

Question

$The coefficient of x^{5} in the expansion of (1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30} is$

Moderate

A

$^{51} C_{5}$
B

$^{9} C_{5}$
C

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

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

Solution

$coefficient of x^{5} i n (1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30} = coefficientof x^{5} i n (1 + x)^{21} [\frac{(1 + x)^{10} - 1}{(1 + x) - 1}] = coefficient of x^{6} i n (1 + x)^{31} - (1 + x)^{21}$

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