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JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Binomial Theorem

The coefficient of

x^{18}

in the expansion of

(x^{4} - \frac{1}{x ^{3}})^{15}

is

Answer: 5005

Solution

Using the binomial theorem:

T_{r + 1} =^{15} C_{r} (x^{4})^{15 - r} (- \frac{1}{x ^{3}})^{r} =^{15} C_{r} (- 1)^{r} x^{60 - 4 r - 3 r} =^{15} C_{r} (- 1)^{r} x^{60 - 7 r}

For the coefficient of

x^{18}

:

60 - 7 r = 18

7 r = 42

r = 6

The coefficient is:

^{15} C_{6} (- 1)^{6} =^{15} C_{6} = \frac{15 !}{6 ! \cdot 9 !} = 5005

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Integer answer

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