Find the coefficient of `x^(50)` after simplifying and collecting the like terms in the expansion of `(1+x)^(1000)+x(1+x)^(999)+x^2(1+x)^(998)++x^(1000)dot`

Explanation

We know that th above term is geometric series with common ratio` x/(1+x)`

its sum is `(a(1-r^n))/(1-r)`

=`[(1+x)^100-(x^100)/(1+x)]/[(1+x-x)/(1+x)]`

=`(1+x)^(1001)-(x)^(1001)`

Hence coefficient of` x^(50)` is

`=1001 _(C _50)`