Question

The coefficient of $x^r(0\le r\le n-1)$ in the expansion of
$E={(x+2)}^{n-1}+{(x+2)}^{n-2}(x+1)+{(x+2)}^{n-3}{(x+1)}^2+....+{(x+1)}^{n-1}$

• A. ${{}^{n}C}_r$
• B. ${{}^{n}C}_r(2^{n-r}-1)$
• C. ${{}^{n}C}_r{2}^{n-r}$
• D. none of these

Answer 1

The correct option is B ${{}^{n}C}_r(2^{n-r}-1)$

Sequence E is in G.P
Therefore, sum of G.P $E={(x+2)}^n-{(x+1)}^n$
Now, coefficient of $x^r$ is ${{}^{n}C}_r{2}^{n-r}-{{}^{n}C}_r={{}^{n}C}_r(2^{n-r}-1)$