A) \[\frac{n+1}{n!}\]
B) \[\frac{{{(n+1)}^{n}}}{(n-1)!}\]
C) \[\frac{{{(n-1)}^{n}}}{n!}\]
D) \[\frac{{{(n+1)}^{n}}}{n!}\]
E) \[\frac{n-1}{n!}\]
Correct Answer: D
Solution :
\[\left( 1+\frac{{{C}_{1}}}{{{C}_{0}}} \right)\left( 1+\frac{{{C}_{2}}}{{{C}_{1}}} \right)\left( 1+\frac{{{C}_{3}}}{{{C}_{2}}} \right)....\left( 1+\frac{{{C}_{n}}}{{{C}_{n-1}}} \right)\] \[=\left( 1+\frac{n}{1} \right)\left( 1+\frac{n(n-1)}{2n} \right).....\left( 1+\frac{1}{n} \right)\] \[=\left( \frac{1+n}{1} \right)\left( \frac{1+n}{2} \right).....\left( \frac{1+n}{n} \right)\] \[=\frac{{{(1+n)}^{n}}}{n!}\]
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