The coefficient of x 5 in the expansion of 1- x 21-1- x 22-1- x 30 isA- 51 C5- 9 C 5- 31 C 6-21 C 6D- 30 C 5-20 C 5

The correct option is C ^31C_6 ^21C_6(1+x)^21 + (1+x)^22 +...+(1+x)^30 = (1+x)^21[(1+x)^10 1(1+x) 1] = 1x[(1+x)^31 (1+x)^21] ⇒ Coefficient of x^5 in the given ...

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Standard XII

Mathematics

Binomial Coefficients

The coefficie...

Question

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

^{51} C_{5}

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^{9} C_{5}

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

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^{30} C_{5} -^{20} C_{5}

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Solution

The correct option is C $^{31} C_{6} -^{21} C_{6}$
$(1 + x)^{21} + (1 + x)^{22} + . . . + (1 + x)^{30}$
$= (1 + x)^{21} [\frac{(1 + x)^{10} - 1}{(1 + x) - 1}] = \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]$
$\Rightarrow$ Coefficient of $x^{5}$ in the given expression is equal to coefficient of $x^{6}$ in $[(1 + x)^{31} - (1 + x)^{21}]$ , which is given by $^{31} C_{6} -^{21} C_{6}$

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Similar questions

Q. 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

Q. The coefficient of

x^{5}

in the expansion of

{(1 + x)}^{21} + {(1 + x)}^{22} + \dots + {(1 + x)}^{30}

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

Q. The coefficient of

x^{5}

in the expansion of

(1 + x)^{21} + (1 + x)^{22} + . . . . . + (1 + x)^{30}

Q. The coefficient of

x^{5}

in the expansion of

(1 + x)^{21} + (1 + x)^{22} + . . . + (1 + x)^{30}

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Binomial Coefficients

Standard XII Mathematics

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The coefficient of x5 in the expansion of (1+x)21+(1+x)22+...+(1+x)30 is

The correct option is C 31C6−21C6(1+x)21+(1+x)22+...+(1+x)30 =(1+x)21[(1+x)10−1(1+x)−1]=1x[(1+x)31−(1+x)21] ⇒ Coefficient of x5 in the given expression is equal to coefficient of x6 in [(1+x)31−(1+x)21], which is given by 31C6−21C6

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