Chapter 12: Q 29. (page 836)

The coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10}$

Short Answer

Expert verified

The coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10}$ is $17010$ .

Step by step solution

Step 1. Definition

Based on the expansion of ${(x + a)}^{n}$ , the term containing $x^{j}$ is $(\begin{matrix} n \\ n - j \end{matrix}) a^{n - j} x^{j}$ .

Step 2. Explanation

The term containing $x^{6}$ in the expansion of ${(x + 3)}^{10}$ is:

localid="1647190158816" $= (\begin{matrix} 10 \\ 10 - 6 \end{matrix}) {(3)}^{10 - 6} x^{6}$

$= \frac{10!}{4! (10 - 4)!} {(3)}^{4} x^{6}$

$= \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!} (81) x^{6}$

$= (210) (81) x^{6}$

$= 17010 x^{6}$

So, the coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10} = 17010$ .

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The coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10}$

Short Answer

Step by step solution

Step 1. Definition

Step 2. Explanation

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The coefficient of $x^{6}$ in the expansion of ${(x + 3)}^{10}$