Find the coefficient of x50  in the expansion :

 

(1+x)1000 + 2x(1+x)999 +3x2(1+x)998+…………………..+1001x1000

Let S=(1+x)1000+2x(1+x)99+3x2(1+x)998+............+1000(1+x)x999+1001x1000.....(1)xS1+x=x(1+x)999+2x2(1+x)998+......1000x1000+1001x10011+x.........(2)Now (1)-(2) ,we getS-xS1+x=(1+x)1000+x(1+x)999+x2(1+x)998+......x1000-1001x10011+xS1+x=(1+x)10001-(x1+x)10011-x1+x-1001x10011+xS1+x=(1+x)1001-x1001-1001x10011+x  S=(1+x)1002-x1001(1+x)-1001x1001S=(1+x)1002-1002x1001-x1002Now coefficient of x50in S = coefficient of x50in (1+x)1002=1002C50

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