The coefficient of x 10 in the expansion of -1- x 21- x -8 isA- 400B- 576- 470D- 476

The correct option is D 476We have, [1+x^2(1 x)]^8 =(1+x^2 x^3)^8 nbsp;= ∑_k_1+k_2+k_3 = 88!k_1!k_2!k_3!(1)^k_1(x^2)^k_2( x^3)^k_3 nbsp;= nbsp;∑_k_1+ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Multinomial Expansion

The coefficie...

Question

The coefficient of $x^{10}$ in the expansion of $[1 + x^{2} (1 - x)]^{8}$ is

470

No worries! We‘ve got your back. Try BYJU‘S free classes today!

400

No worries! We‘ve got your back. Try BYJU‘S free classes today!

476

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

576

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $476$
We have,
$[1 + x^{2} (1 - x)]^{8} = (1 + x^{2} - x^{3})^{8} = \sum k_{1} + k_{2} + k_{3} = 8 \frac{8!}{k_{1}! k_{2}! k_{3}!} (1)^{k_{1}} (x^{2})^{k_{2}} (- x^{3})^{k_{3}} = \sum k_{1} + k_{2} + k_{3} = 8 \frac{8!}{k_{1}! k_{2}! k_{3}!} (- 1)^{k_{3}} x^{2 k_{2} + 3 k_{3}}$

Coefficient of $x^{10}$ , so
$2 k_{2} + 3 k_{3} = 10 k_{1} + k_{2} + k_{3} = 8$

So,
$(1) k_{1} = 4, k_{2} = 2, k_{3} = 2 (2) k_{1} = 3, k_{2} = 5, k_{3} = 0$

Therefore, the required coefficient
$= \frac{8!}{4! 2! 2!} + \frac{8!}{3! 5! 0!} = \frac{8 \cdot 7 \cdot 6 \cdot 5}{4} + \frac{8 \cdot 7 \cdot 6}{6} = 420 + 56 = 476$

Suggest Corrections

Similar questions

The coefficient of $x^{9}$ in the expansion of $1 + x) ((1 + x^{2})$ $(1 + x^{3})$ $\dots$ $(1 + x^{100})$ is

Q. The coefficient of

x^{10}

in the expansion of

(1 + x)^{2} {(1 + x^{2})}^{3} {(1 + x^{3})}^{4}

is equal to:

Q. The coefficient of

x^{10}

in the expansion of

(1 + x)^{2} (1 + x^{2})^{3} (1 + x^{3})^{4}

is qual to:

Q. The coefficient of

x^{10}

in the expansion of

(1 + x)^{2} (1 + x^{2})^{3} (1 + x^{3})^{4}

is equal to :

Q. The coefficient of

x^{7}

in the expression

(1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10}

is :

Join BYJU'S Learning Program

Explore more

Multinomial Expansion

Standard XII Mathematics

Join BYJU'S Learning Program

The coefficient of x10 in the expansion of [1+x2(1−x)]8 is

The coefficient of $x^{10}$ in the expansion of $[1 + x^{2} (1 - x)]^{8}$ is