We have to find the coefficient of x12 in (x3 + x4 + x5 + x6 + …)3
(x3 + x4 + x5 + x6 + …)3 = [ x3 (1 + x1 + x2 + x3 + x4 + x5 + …)]3
= [ x9 (1 + x1 + x2 + x3 + x4 + x5 + …)3]
Now, in remaining term we need to find coefficient of x3. Because x12 = x9 × x3.
Now, [ x9 (1 + x1 + x2 + x3 +…… )3] = \({x^9}{\left( {\frac{1}{{1 - x}}} \right)^3} = {x^9}{\left( {1 - x} \right)^{ - 3}}\)
Now, by using binomial coefficient here,
x9 (1 – x)-3 = x9 (1 + 3x + 6x2 + 10x3 + …) = x9 + 3x10 + 6x11 + 10x12 + …
Here, it is clearly shown that, coefficient of x12 is 10.
