combinatorics

Question

coefficient of x^20 in (1+x+x^2+......................)^2 is?

Answer 1

(1 + x + x²  + x³ .......)²   =  (1 - x)^-2 

So coefficient of x²⁰  in (1 + x + x²  + x³ .......)²       =      (1 - x)^-2   

So it is found using multinomial theorem as : 

We know :

Coefficient of x^r in ( 1 - x ) ^-n    =  ^n-1+rC_r

So here n = 2 and r  = 20 , so we find this as :

Coefficient of x²⁰  in  (1 - x)^-2  , we have : ^{2 - 1 + 20}C₂₀

                                                      =     ²¹C₂₀

                                                      =     21

Hence 21 is the correct answer..