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.