Find coefficient of $x^{4}$ in $(1 + x + x^{3} + x^{4})^{10}$ .

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Solution

$(1 + x + x^{3} + x^{4})^{10}$
To find coeff.of $x^{4}$
$= \sum p, q \geq 0 \frac{10! (- 1)^{q}}{p! . q! (10 - p - q)!} (x^{3})^{p} . (x)^{2}$
For $p = 1, q = 1$
$= \frac{10! (- 1) x^{4}}{1! (10 - 2)!} = \frac{10!}{8!} = 10 \times 9 = 90... (1)$
For $p = 4, q = 0$
$\sum p, q \geq 0 \frac{10! (- 1)^{q}}{p! . q! (10 - p - q)!} (x)^{p} (x^{3})^{q}$
$= \frac{10!}{4! (10 - 4)!} x^{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2} x^{4} = 210 . . . . (2)$

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Find coefficient of x4 in (1+x+x3+x4)10.

(1+x+x3+x4)10To find coeff.of x4=∑p,q≥010!(−1)qp!.q!(10−p−q)!(x3)p.(x)2For p=1,q=1=10!(−1)x41!(10−2)!=10!8!=10×9=90...(1)For p=4,q=0∑p,q≥010!(−1)qp!.q!(10−p−q)!(x)p(x3)q=10!4!(10−4)!x4=10×9×8×7×6!4!×6!=10×9×8×74×3×2x4=210....(2)

Find coefficient of $x^{4}$ in $(1 + x + x^{3} + x^{4})^{10}$ .