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JEE Main 202311 AprilMorning Shift
Exam
JEE Main
Year
2023
Date
11 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2023 · 11 April · Morning Shift · Mathematics · Binomial Theorem

The mean of the coefficients of x, x ^ 2 , , x ^ 7 in the binomial...

The mean of the coefficients of x,x2,……,x7x, {x}^{2}, \ldots \ldots , {x}^{7} in the binomial expression of (2+x)9( 2 + x{)}^{9} is _________

Correct answer

0

Step-by-step solution

We know that.

Binomial expansion of (2+x)9isgivenby,</p><p>{\left(2 + x\right)}^{9} is given by,</p><p>{\left(2 + x\right)}^{9} = C0 9 {x}^{0}{2}^{9} + C1 9 {x}^{1}{2}^{8} + . . . . . . . . . . . C9 9 {x}^{9}{2}^{0}</p><p>Nowput</p><p>Now putx = 1weget,</p><p>we get,</p><p>{\left(2 + 1\right)}^{9} = C0 9 {2}^{9} + C1 9 {2}^{8} + + C2 9 {2}^{7} + . . . . . . . . . . . C9 9 {2}^{0}</p><p></p><p>⇒ {3}^{9} = C0 9 {2}^{9} + C1 9 {2}^{8} + + C2 9 {2}^{7} + . . . . . . . . . . . C9 9 {2}^{0}</p><p>So,sumofcoefficientof</p><p>So, sum of coefficient ofx, {x}^{2}, {x}^{3}, . . . . . . . . . , {x}^{7}willbe,</p><p>will be,</p><p>⇒ C1 9 {2}^{8} + C2 9 {2}^{7} + . . . . . . . . . . . C7 9 {2}^{2} = {3}^{9} - C0 9 {2}^{9} - C8 9 {2}^{1} - C9 9 {2}^{0}</p><p></p><p>⇒ C1 9 {2}^{8} + C2 9 {2}^{7} + . . . . . . . . . . . C7 9 {2}^{2} = {3}^{9} - {2}^{9} - 18 - 1</p><p></p><p>⇒ C1 9 {2}^{8} + C2 9 {2}^{7} + . . . . . . . . . . . C7 9 {2}^{2} = 19683 - 512 - 19</p><p></p><p>⇒ C1 9 {2}^{8} + C2 9 {2}^{7} + . . . . . . . . . . . C7 9 {2}^{2} = 19152</p><p>Nowmeanisgivenby</p><p>Now mean is given by\dfrac{19152}{7} = 2736

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