`(1+2x)^(6)(1-x)^(7)` के गुणनफल के प्रसार में `x^(5)` का गुणांक ज्ञात कीजिए |

Explanation

यहाँ

`(1+2x)^(6)``= ""^6C_(0) 1^(6-0)(2x)^0``+""^(6)C_(1) 1^(6-1)(2x)^(1)``+""^(6)C_(2) 1^(6-2)(2x)^(2)``+""^(6)C_(3) 1^(6-3)(2x)^(3)``+""^(6)C_(4) 1^(6-4)(2x)^(4)``+""^(6)C_(5) 1^(6-5)(2x)^(5)``+""^(6)C_(6) 1^(6-6)(2x)^(6)``...(1)`

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

पुन: `(1-x)^(7)``=1-""^(7)C_(1)x``+""^(7)C_(2)x^(2)``-""^(7)C_(3)x^(3)``+""^(7)C_(4)x^(4)``-""^(7)C_(5)x^(5)``+""^(7)C_(6)x^(6)``-""^(7)C_(7)x^(7)`

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

अब, `(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)`

`:.` गुणनफल में `x^(5)` का अभीष्ट गुणांक

`= 1xx(-21)``+12xx35``+60xx(-35)``+160xx21``+240xx(-7)``+192xx1`

`= -21``+420``-2100``+3360``-1680``+192``=171`