The coefficient of x ^ 18 in the expansion of ( x ^ 4 - 1 x ^ 3 ) ^ 15 is ____________

Correct answer: 5005 Given, Binomial expression ( x ^ 4 - 1 x ^ 3 ) ^ 15 Now general term is given by, T _ r + 1 = Cr 15 ( x ^ 4 ) ^ 15 - r (- 1 x ^ 3 ) ^ r ⇒ T _ r + 1 = Cr 15 x ^ 60 - 7r ( - 1 ) ^ r For coefficient of x ^ 18 , equate 60 - 7r = 18 ⇒ r = 6 Coefficient of x ^ 18 = (- 1) ^ 6 · C6 15 = 5005

JEE Main 20236 AprilMorning Shift

Exam: JEE Main
Year: 2023
Date: 6 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2023 · 6 April · Morning Shift · Mathematics · Binomial Theorem

The coefficient of x ^ 18 in the expansion of ( x ^ 4 - 1 x ^ 3 ) ^...

The coefficient of

{x}^{18}

in the expansion of

{\left({x}^{4} - \dfrac{1}{{x}^{3}}\right)}

^{15}

is

____________

Correct answer

5005

Step-by-step solution

Given,
Binomial expression ${\left({x}^{4} - \dfrac{1}{{x}^{3}}\right)}^{15}$
Now general term is given by,
${T}_{r + 1} = Cr 15 {\left({x}^{4}\right)}^{15 - r}{\left(- \dfrac{1}{{x}^{3}}\right)}^{r}$
$⇒ {T}_{r + 1} = Cr 15 {x}^{60 - 7r} ( - 1{)}^{r}$
For coefficient of ${x}^{18},$ equate $60 - 7r = 18$
$⇒ r = 6$
Coefficient of ${x}^{18} = {\left(- 1\right)}^{6} \cdot C6 15 = 5005$

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