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Find the coefficient of x^4 in the expansion of (1+x+ x^2+x^3)

Question

Find the coefficient of $x^{4}$ in the expansion of $(1 + x + x^{2} + x^{3})^{11}$ .

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Find the coefficient of

x^{4}

in the expansion of

(1 + x + x^{2} + x^{3})^{11}

10 mins ago

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10 mins ago

Text solutionVerified

(1 + x + x^{2} + x^{3})^{11}

((1 + x) (1 + x^{2}))^{11}

(1 + x)^{11} (1 + x^{2})^{11}

[1 1_{C_{0}} (x^{0})

+ 1 1_{C_{1}} (x^{1})

+ 1 1_{C_{2}} (x^{2})

+ ..... + 1 1_{C_{1} 1} (x^{11})]

[1 1_{C_{0}} (x^{2})^{0}

+ 1 1_{C_{1}} (x^{2})^{1}

+ 1 1_{C_{2}} (x^{2})^{2}

+ ..... + 1 1_{C_{1} 1} (x^{2})^{11}]

Coefficient of

x^{4}

1 1_{C_{2}} .1 1_{C_{0}}

+ 1 1_{C_{1}} .1 1_{C_{2}}

+ 1 1_{C_{0}} .1 1_{C_{4}}

= \frac{11 !}{9 ! 2 !} .1

+ \frac{11 !}{10 ! 1 !} .

\frac{11 !}{9 ! 2 !}

+ \frac{11 !}{4 ! 7 !} .1

= 11 \times 5 + 11 \times 11 \times 5 + 110 \times 3 = 55 + 605 + 330 = 990

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Question Text	Find the coefficient of $x^{4}$ in the expansion of $(1 + x + x^{2} + x^{3})^{11}$ .
Answer Type	Text solution:1
Upvotes	150