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Q3.    Find the coefficient of x^5 in the product (1 + 2x)^6 (1 - x)^7 using binomial theorem.

Answers (1)

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First, lets expand both expressions individually,

So,

(1+2x)^6=^6C_0+^6C_1(2x)+^6C_2(2x)^2+^6C_3(2x)^3+^6C_4(2x)^4+^6C_5(2x)^5+^6C_6(2x)^6

(1+2x)^6=^6C_0+2\times^6C_1x+4\times^6C_2x^2+8\times^6C_3x^3+16\times^6C_4x^4+32\times^6C_5x^5+64\times^6C_6x^6

(1+2x)^6=1+12x+60x^2+160x^3+240x^4+192x^5+64x^6

And 

(1-x)^7=^7C_0-^7C_1x+^7C_2x^2-^7C_3x^3+^7C_4x^4-^7C_5x^5+^7C_6x^6-^7C_7x^7

(1-x)^7=1-7x+21x^2-35x^3+35x^4-21x^5+7x^6-x^7

Now,

(1 + 2x)^6 (1 - x)^7=(1+12x+60x^2+160x^3+240x^4+192x^5+64x^6)(1-7x+21x^2-35x^3+35x^4-21x^5+7x^6-x^7)

Now, for the coefficient of x^5, we multiply and add those terms whose product gives x^5.So,

The term which contain x^5are,

\Rightarrow (1)(-21x^5)+(12x)(35x^4)+(60x^2)(-35x^3)+(160x^3)(21x^2)+(240x^4)(-7x)+(192x^5)(1)

\Rightarrow 171x^5

Hence the coefficient of x^5 is 171.

Posted by

Pankaj Sanodiya

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