Chapter 12: Q. 32 (page 839)

Find the coefficient of $x^{7}$ in the expansion of ${(x + 2)}^{9} .$

Short Answer

Expert verified

The coefficient of $x^{7}$ is $144$ .

Step by step solution

Given information:

Consider the expression ${(x + 2)}^{9}$ .

To find the coefficient of $x^{7}$ .

Formulas :

Formula 1: To find a particular term in an expression:

${(x + a)}^{n} = (\begin{matrix} n \\ n - j \end{matrix}) a^{(n - j)} x^{j}$

Formula 2: Binomial Coefficient:

$(\begin{matrix} n \\ j \end{matrix}) = \frac{n!}{j! \times (n - j)!}$

Step 3:

On comparing ${(x + 2)}^{9}$ with ${(x + a)}^{n}$ , we get,

$x = x, a = 2, n = 9 .$

To find the coefficient of $x^{7}$ ,role="math" localid="1647343175949" $\Rightarrow j = 7$

Substituting the values in the formula 1:

${(x + 2)}^{9} = (\begin{matrix} 9 \\ 9 - 7 \end{matrix}) {(2)}^{(9 - 7)} x^{7}$

Simplification:

${(x + 2)}^{9} = (\begin{matrix} 9 \\ 2 \end{matrix}) {(2)}^{2} {(x)}^{7}$

$= 36 \times 4 x^{7}$ $[u s e f o r m u l a 2 : (\begin{matrix} 9 \\ 2 \end{matrix}) = \frac{9!}{2! (9 - 2)!} = \frac{9!}{2! \times 7!} = \frac{9 \times 8 \times 7!}{(2 \times 1) \times 7!} = \frac{9 \times 8}{2} = 36]$

role="math" localid="1647059684716" $= 144 x^{7}$

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Find the coefficient of $x^{7}$ in the expansion of ${(x + 2)}^{9} .$

Short Answer

Step by step solution

Given information:

Formulas :

Step 3:

Substituting the values in the formula 1:

Simplification:

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Short Answer

Step by step solution

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Find the coefficient of $x^{7}$ in the expansion of ${(x + 2)}^{9} .$