Question Number 104134 by bemath last updated on 19/Jul/20

what is the coefficient x^(15) in the expansion of x^6 (1−x)^(11)

Answered by bobhans last updated on 19/Jul/20

εx^6 (1−x)^(11) = x^6 Σ_(n = 0) ^(11) C _n^(11) (1)^n (−x)^(11−n) = x^6 {...+ C _2^(11) (−x)^(11−2) + ... } = x^6 {...+ ((11.10)/(2.1)) (−x^9 )+...} so coefficient x^(15) is −55 . ■

Answered by mathmax by abdo last updated on 19/Jul/20

x^6 (1−x)^(11) =−x^6 (x−1)^(11) =−x^6 Σ_(k=0) ^(11 ) C_(11) ^k x^k (−1)^(11−k) =x^6 Σ_(k=0) ^(11) C_(11) ^k (−1)^k x^k =Σ_(k=0) ^(11) (−1)^(k ) C_(11) ^k x^(k+6) we get the coefficient of x^(15) ifk+6 =15 ⇒k =9 so tbe coefficent is λ =(−1)^9 C_(11) ^9 =−((11!)/(9!2!)) =−((11×10)/2) =−55.