Find the coefficient of `x^5` in the expansioin of the product `(1+2x)^6(1-x)^7dot`

Explanation

We know that

`(a+b)^n=^nC_0a^n+^nC_1a^(n-1)b^1+........+^nC_(n-1)a^1b^(n-1)+^nC_nb^n`

Hence,
`(a+b)^6=^6C_0a^6+^6C_1a^(5)b^1+^6C_2a^(4)b^2+^6C_3a^(3)b^3+^6C_4a^(2)b^4+^6C_5a^(1)b^5+^6C_6b^6`

` " " " " = a^6+(6!)/(1!(6-1)!)a^(5)b^1+(6!)/(2!(6-2)!)a^(4)b^2+(6!)/(3!(6-3)!)a^(3)b^3+(6!)/(4!(6-4)!)a^(2)b^4+(6!)/(5!(6-5)!)a^(1)b^5+b^6`

` " " " " =a^6+(6!)/(1xx5!)a^(5)b^1+(6!)/(2!xx4!)a^(4)b^2+(6!)/(3!3!)a^(3)b^3+(6!)/(4!2!)a^(2)b^4+(6!)/(5!xx1)a^(1)b^5+b^6`

` " " " " =a^6+(6xx5!)/(5!)a^(5)b^1+(6xx5xx4!)/(2xx4!)a^(4)b^2+(6xx5xx4xx3!)/(3xx2xx1xx3!)a^(3)b^3+(6xx5xx4!)/(4!xx2xx1)a^(2)b^4+(6xx5!)/(5!)a^(1)b^5+b^6 `

` " " " " =a^6+6a^(5)b+15a^(4)b^2+20a^(3)b^3+15a^(2)b^4+6ab^5+b^6`

Putting `a=1` & `b=2x`

`(1+2x)^6=1^6+6(1)^(5)(2x)+15(1)^(4)(2x)^2+20(1)^(3)(2x)^3+15(1)^(2)(2x)^4+6(1)(2x)^5+(2x)^6`

` " " " " =1+6(2x)+15(4x^2)+20(8x^3)+15(16x^4)+6(32x^5)+64x^6`

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

Similarly,

We know that

`(a+b)^n=^nC_0a^n+^nC_1a^(n-1)b^1+........+^nC_(n-1)a^1b^(n-1)+^nC_nb^n`

Hence,
`(a+b)^7=^7C_0a^7+^7C_1a^(6)b^1+^7C_2a^(5)b^2+^7C_3a^(4)b^3+^7C_4a^(3)b^4+^7C_5a^(2)b^5+^7C_5a^(1)b^6+^7C_6b^7`

` " " " " = a^7+(7!)/(1!(7-1)!)a^(6)b^1+(7!)/(2!(7-2)!)a^(5)b^2+(7!)/(3!(7-3)!)a^(4)b^3+(7!)/(4!(7-4)!)a^(3)b^4+(7!)/(5!(7-5)!)a^(2)b^5+(7!)/(5!(7-6)!)a^(1)b^6+b^7`

` " " " " =a^7+(7!)/(1xx6!)a^(6)b^1+(7!)/(2!xx5!)a^(5)b^2+(7!)/(3!4!)a^(4)b^3+(7!)/(4!3!)a^(3)b^4+(7!)/(5!xx2!)a^(2)b^5+(7!)/(6!xx1!)a^(1)b^6+b^7`

` " " " " =a^7+(7xx6!)/(6!)a^(6)b^1+(7xx6xx5!)/(2xx5!)a^(5)b^2+(7xx6xx5xx4!)/(3xx2xx1xx4!)a^(4)b^3+(7xx6xx5xx4!)/(4!xx3xx2xx1)a^(3)b^4+(7xx6xx5!)/(5!xx2)a^(2)b^5+(7xx6!)/(6!)a^(1)b^6+b^7 `

` " " " " =a^7+7a^(6)b+21a^(5)b^2+35a^(4)b^3+35a^(3)b^4+21a^2b^5+7ab^6+b^7`

Putting `a=1` & `b=-x`

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




Coefficient of `x^5` is `171`