- Home
- Standard 11
- Mathematics
The coefficient of $x^r (0 \le r \le n - 1)$ in the expression :
$(x + 2)^{n-1} + (x + 2)^{n-2}. (x + 1) + (x + 2)^{n-3} . (x + 1)^2; + ...... + (x + 1)^{n-1}$ is :
$^nC_r (2^r - 1)$
$^nC_r (2^{n-r} - 1)$
$^nC_r (2^r + 1)$
$^nC_r (2^{n-r} + 1)$
Solution
$E = (x + 2)^{n – 1}$ $\left[ {\,1\,\, + \,\,\frac{{x\,\, + \,\,1}}{{x\,\, + \,\,2}}\,\, + \,\,{{\left( {\frac{{x\,\, + \,\,1}}{{x\,\, + \,\,2}}} \right)}^2}\,\, + \,\,……\,\, + \,\,{{\left( {\frac{{x\,\, + \,\,1}}{{x\,\, + \,\,2}}} \right)}^{n\, – \,1}}\,} \right]$
$= (x + 2)^{n – 1}$ $\left[ {\frac{{1\,\, – \,\,{{\left( {{\textstyle{{x\,\, + \,\,1} \over {x\,\, + \,\,2}}}} \right)}^n}}}{{1\,\, – \,\,{\textstyle{{x\,\, + \,\,1} \over {x\,\, + \,\,2}}}}}} \right]$
$= (x + 2)^n$ $\left[ {\frac{{{{(x\,\, + \,\,2)}^n}\,\, – \,\,{{(x\,\, + \,\,1)}^n}}}{{{{(x\,\, + \,\,2)}^n}}}} \right]$
$= (x + 2)^n – (x + 1)^n$
Now co-efficient of $x^r$ in $\left[ {\,{{(2\,\, + \,\,x)}^n}\,\, – \,\,{{(1\,\, + \,\,x)}^n}\,} \right] = ^nC_r 2^{n – r} – ^nC_r = ^nC_r (2^{n – r} – 1)$
