# The Coefficient of X5 in the Expansion of ( 1 + X ) 21 + ( 1 + X ) 22 + . . . + ( 1 + X ) 30(A) 51c5 (B) 9c5 (C) 31c6 − 21c6 (D) 30c5 + 20c5 - Mathematics

MCQ

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

• 51C5

•  9C5

•  31C6 − 21C6

•  30C5 + 20C5

#### Solution

31C6 − 21C6

$\text{ We have } \left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . \left( 1 + x \right)^{30}$
$= \left( 1 + x \right)^{21} \left[ \frac{\left( 1 + x \right)^{10} - 1}{\left( 1 + x \right) - 1} \right]$
$= \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]$
$\text{ Coefficient of } x^5 \text{ in the given expansion = Coefficient of } x^5 \text{ in } \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]$
$= \text{ Coefficient of } x^6 \text{ in }\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]$
$=^{31} C_6 -^{21} C_6$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Q 25 | Page 48