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Coefficient of $x^{n-6}$ in the expansion $n\left[ {x - \left( {\frac{{^n{C_0}{ + ^n}{C_1}}}{{^n{C_0}}}} \right)} \right]\left[ {\frac{x}{2} - \left( {\frac{{^n{C_1}{ + ^n}{C_2}}}{{^n{C_1}}}} \right)} \right]\left[ {\frac{x}{3} - \left( {\frac{{^n{C_2}{ + ^n}{C_3}}}{{^n{C_2}}}} \right)} \right].....$ $ \left[ {\frac{x}{n} - \left( {\frac{{^n{C_{n - 1}}{ + ^n}{C_n}}}{{^n{C_{n - 1}}}}} \right)} \right]$ is equal to (where $n = n . (n -1) . (n -2).... 3.2.1$ )
$^n{C_6}{\left( {n + 1} \right)^6}$
$^n{C_6}.{n^6}$
$^n{C_6}{\left( {n + 2} \right)^6}$
$^n{C_5}{\left( {n + 1} \right)^5}$
Solution
$n!\left[ {x – \frac{{\left( {n + 1} \right)}}{1}} \right]\left[ {\frac{x}{2} – \frac{{\left( {n + 1} \right)}}{2}} \right]$
$\left[\frac{\mathrm{x}}{3}-\frac{(\mathrm{n}+1)}{3}\right] \ldots\left[\frac{\mathrm{x}}{\mathrm{n}}-\frac{(\mathrm{n}+1)}{\mathrm{n}}\right]=[\mathrm{x}-(\mathrm{n}+1)]^{\mathrm{n}}$
Coefficient of $x^{n-6}$ is $^{n} C_{6}(n+1)^{6}$
