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Generating function

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2 2 votes

The simplest way should be, 

to find coefficient of x^4 in the expansion of (1+x-2x^2)^7 is....

    = (1 + x-2x2) can be broken as  ( 1+2x) * (1-x)...

    =coefficient of x4 in  ((1+2x) (1-x)) 7

                             = (1+2x)7 (1-x)7  ( say= (A)7(B)7 )

( so we need to find coeff. of x4 from A and x0 from B and x3 from A and x from B ..........................so on till (sum of coeff. of both = 4))

                          = 7C4 (1)0 (2)4 * 7C0 (1)4 (-1)0  + 7C3 (1)1 (2)3 * 7C1(1)3(-1)1+ 7C2 (1)2 (2)2 * 7C2 (1)2 (-1)2  + 7C1 (1)3 (2)1 * 7C3 (1)1 (-1)3 + 7C0 (1)4 (2)0 * 7C4 (1)0 (-1)4

                       = 35*16 + 35 * -56 + 21*4*21 +  35*-14 +35

                       = -91 ANSWER.

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1 1 vote

co-eff X4 lies in     $\binom{7}{2}$ (x-2x2)2 + $\binom{7}{3}$ (x-2x2)3 + $\binom{7}{4}$ (x-2x2)4 

                              $\binom{7}{2}$ x2(1-2x)2  + $\binom{7}{3}$ x3(1-2x)3  + $\binom{7}{4}$ x4(1-2x)4 

                              $\binom{7}{2}$ *4 + $\binom{7}{3}$ *-6 + $\binom{7}{4}$

                               84-210+35=-91

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