Question

What is the coefficient of x⁴ in the expansion of (1 + x)¹¹ + (1 + x)¹² + ..... + (1 + x)²⁰?

Answer 1

(1 + x)¹¹ + (1 + x)¹² + ..... + (1 + x)²⁰ 

\(\big[\because This\, is \,the \,sum\, of\, a\, G.P.\, with\, 10\, terms\, whose\, first\, term \,(1+ x)^{11}\, and\, common\, ratio\, (1 + x ) \big]\)

∴ Coeff. of x⁴ in the expansion of \(\frac{1}{x}\)[ (1+x)²¹ - (1+x)¹¹]

= Coeff. of x⁵ in [(1 + x) ²¹ – (1 + x) ¹¹] 

= ²¹C₅ – ¹¹C₅