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

$$\mathrm{what}\mathrm{is}\mathrm{coefficient}{\mathrm{x}}^5\mathrm{in}\mathrm{expansion}$$$${(1+{\mathrm{x}}^2)}^5\times {(1+\mathrm{x})}^4$$

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

$$\underset{k=0}{\sum ^5}{C}_k^5{x}^{2k}\underset{r=0}{\sum ^4}{C}_r^4{x}^r$$$$2k+r=5$$$$\Rightarrow k=1,r=3\Rightarrow C_1^5{C}_3^4$$$$\Rightarrow k=2,r=1\Rightarrow C_2^5{C}_1^4$$$$C_1^5{C}_3^4+C_2^5{C}_1^4=20+40=60\Rightarrow answer$$

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

$$\mathrm{thank}\mathrm{you}$$

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

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

$$(1+5c_1{x}^2+5c_2{x}^4+5c_3{x}^6+5c_4{x}^8+x^{10})(1+4c_{1}x+4_{}{c}_2{x}^2+4c_3{x}^3+x^4)$$$$5c_1{x}^2\times 4c_3{x}^3+5c_2{x}^4\times 4c_{1}x$$$$=x^5(5\times 4+10\times 4)$$$$=60x^5$$

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

$$\mathrm{thank}\mathrm{you}$$

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

$$mostwelcome$$

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