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JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Binomial Theorem

The coefficient of

x^{7}

in

(1 - x + 2 x^{3})^{10}

is ______.

Answer: 960

Solution

In the multinomial expansion of

(1 - x + 2 x^{3})^{10}

a general term is

\frac{10 !}{r _{1} ! r _{2} ! r _{3} !} (1)^{r_{1}} (- 1)^{r_{2}} (2)^{r_{3}} x^{r_{2} + 3 r_{3}}

with

r_{1} + r_{2} + r_{3} = 10

. The power of

x

is

7

when

r_{2} + 3 r_{3} = 7

, giving

(r_{1}, r_{2}, r_{3}) = (3, 7, 0), (5, 4, 1), (7, 1, 2)

. Their contributions are

\frac{10 !}{3 ! 7 !} (- 1)^{7} = - 120

,

\frac{10 !}{5 ! 4 ! 1 !} (2) = 2520

, and

\frac{10 !}{7 ! 1 ! 2 !} (- 1) (4) = - 1440

. Adding these, the coefficient of

x^{7}

is

- 120 + 2520 - 1440 = 960

.

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Integer answer

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