Let given expansion be $S = \left(1+x\right)^{1000} + x \left(1+x\right)^{999} + x^{2} \left(1+x\right)^{998} + ... + ...+ x^{1000}$
Put 1 + x = t
$ S= t^{1000} + xt^{999} + x^{2} \left(t\right)^{998} + ...+x^{1000}$
This is a G.P with common ratio $ \frac{x}{t} $$S = \frac{t^{1000} \left[ 1 - \left(\frac{x}{t}\right)^{1001}\right]}{1- \frac{x}{t}} $$=\frac{\left(1+x\right)^{1000} \left[1- \left(\frac{x}{1+x}\right)^{1001}\right]}{ 1 - \frac{x}{1+x}}$$ = \frac{\left(1+x\right)^{1001} \left[\left(1+x\right)^{1001} -x^{1001}\right]}{\left(1+x\right)^{1001}} $$= \left[\left(1+x\right)^{1001} - x^{1001}\right]$
Now coeff of $x^{50}$ in above expansion is equal to coeff of $x^{50}$ in $(1 + x)^{1001}$ which is ${^{1001}C_{50}}$$ = \frac{\left(1001\right)!}{50! \left(951\right)!} $
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
General Term in (1 + x)n is nCr xr
In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .
