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The coefficient of x18 in the expansion of x4frac 1x315... | null (Mathematics) | DivineJEE

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Mathematics

Exam

Mathematics

The coefficient of $x^{18}$ in the expansion of $\left(x^{4}-\frac{1}{x^{3}}\right)^{15}$ is __________.

Correct Answer:

5005

Explanation

\begin{aligned} T_{r+1}= & { }^{15} C_r\left(x^4\right)^{15-r}\left(-\frac{1}{x^3}\right)^r \\& ={ }^{15} C_r(-1)^r x^{60-4 r-3 r} \\ = & { }^{15} C_r(-1)^r x^{60-7 r}\end{aligned}

\begin{aligned} \therefore 60-7 r =18 \\ \Rightarrow 7 r =42 \\ \Rightarrow r =6\end{aligned}

\therefore

The coefficient of

x^{18}

={ }^{15} C_6(-1)^6

\\=\frac{15 \times 14 \times 13 \times 12 \times 11 \times 10}{6 \times 5 \times 4 \times 3 \times 2 \times 1}=5005

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