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`(1 + x^(2))^(4)(1 + x^(3))^(7)(1 + x^(4))^(12)` के विस्तार में `x^(11)` का गुणांक ज्ञात कीजिए । |
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Answer» `because (1 + x)^(n)` में `x^(n)` का गुणांक `.^(n)C_(r)` है | इस प्रकार `(1 + x^(2))^(4) (1 + x^(3))^(7) (1 + x^(4))^(12)` के विस्तार में `x^(0) x^(3) x^(8), x^(2) x^(9) x^(0), x^(4) x^(3) x^(4), x^(8) x^(3) x^(0)` अर्थात `x^(11)` का गुणांक `=.^(4)C_(0) xx .^(7)C_(1)xx .^(12)C_(2)+.^(4)C_(1)xx.^(7)C_(3)xx.^(12)C_(0)+.^(4)C_(2)xx.^(7)C^(1)xx.^(12)C_(1)+.^(4)C_(4)xx .^(7)C_(1)xx .^(12)C_(0)` `=1xx(7!)/(6!)xx(12!)/(2!.10!)+(4!)/(3!)xx(7!)/(3!.4!)xx1+(4!)/(2!.2!)xx(7!)/(6!)xx(12!)/(11!)+1xx(7!)/(6!)xx 1` `=7xx(12xx11)/(2)+(4xx7xx6xx5)/(3xx2)+(4xx3)/(2)xx7xx12+7` `=462 + 140 + 504 + 7` = 1113. |
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