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JEE Main 20209 JanuaryMorning Shift
Exam
JEE Main
Year
2020
Date
9 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2020 · 9 January · Morning Shift · Mathematics · Binomial Theorem

The coefficient of x ^ 4 in the expansion of (1 + x + x ^ 2 ) ^ 10...

The coefficient of x4intheexpansionof{x}^{4} in the expansion of{\left(1 + x + {x}^{2}\right)}^{10} is ________

Correct answer

0

Step-by-step solution

General term 10!α!β!γ!xβ+2γ\dfrac{10 !}{\alpha ! \beta ! \gamma !}{x}^{\beta + 2\gamma}
For coefficient of x4⇒β+2γ=4{x}^{4} ⇒ \beta + 2\gamma = 4
γ=0,β=4,α=6⇒10!6!4!0!=210\gamma = 0, \beta = 4, \alpha = 6 ⇒ \dfrac{10 !}{6 ! 4 ! 0 !} = 210
γ=1,β=2,α=7⇒10!7!2!1!=360\gamma = 1, \beta = 2, \alpha = 7 ⇒ \dfrac{10 !}{7 ! 2 ! 1 !} = 360
γ=2,β=0,α=8⇒10!8!0!2!=45\gamma = 2, \beta = 0, \alpha = 8 ⇒ \dfrac{10 !}{8 ! 0 ! 2 !} = 45
Total =615= 615
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