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JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Binomial Theorem

The coefficient of

x^{4}

in the expansion of

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

is ______.

Answer: 615

Solution

In

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

a general term is

\frac{10 !}{α ! β ! γ !} x^{β + 2 γ}

with

α + β + γ = 10

, and the coefficient of

x^{4}

needs

β + 2 γ = 4

. The possibilities are

(γ, β, α) = (0, 4, 6), (1, 2, 7), (2, 0, 8)

, contributing

\frac{10 !}{6 ! 4 !} = 210

,

\frac{10 !}{7 ! 2 ! 1 !} = 360

and

\frac{10 !}{8 ! 2 !} = 45

. Their total is

210 + 360 + 45 = 615

.

JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Integer answer

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