what-is-coefficient-x-5-in-expansion-1-x-2-5-1-x-4-

Question Number 83166 by john santu last updated on 28/Feb/20

what is coefficient x^{5} in expansion

{(1 + x^{2})}^{5} \times {(1 + x)}^{4}

Commented by mr W last updated on 28/Feb/20

\underset{k = 0}{\sum^{5}} C_{k}^{5} x^{2 k} \underset{r = 0}{\sum^{4}} C_{r}^{4} x^{r}

2 k + r = 5

\Rightarrow k = 1, r = 3 \Rightarrow C_{1}^{5} C_{3}^{4}

\Rightarrow k = 2, r = 1 \Rightarrow C_{2}^{5} C_{1}^{4}

C_{1}^{5} C_{3}^{4} + C_{2}^{5} C_{1}^{4} = 20 + 40 = 60 \Rightarrow a n s w e r

Commented by john santu last updated on 28/Feb/20

thank you

Answered by john santu last updated on 28/Feb/20

Answered by TANMAY PANACEA last updated on 28/Feb/20

(1 + 5 c_{1} x^{2} + 5 c_{2} x^{4} + 5 c_{3} x^{6} + 5 c_{4} x^{8} + x^{10}) (1 + 4 c_{1} x + 4_{} c_{2} x^{2} + 4 c_{3} x^{3} + x^{4})

5 c_{1} x^{2} \times 4 c_{3} x^{3} + 5 c_{2} x^{4} \times 4 c_{1} x

= x^{5} (5 \times 4 + 10 \times 4)

= 60 x^{5}

Commented by john santu last updated on 28/Feb/20

thank you

Commented by TANMAY PANACEA last updated on 28/Feb/20

m o s t w e l c o m e