# Moderate Permutation-Combination Solved QuestionAptitude Discussion

 Q. What is the value of $1×1! + 2×2! + 3×3! + ............ n×n!$; where $n!$ means $n$ factorial or $n(n-1)(n-2)...1$
 ✖ A. $n(n-1)(n-1)!$ ✖ B. ${(n+1)!}/{n(n-1)}$ ✖ C. $(n+1)! - n!$ ✔ D. $(n + 1)! - 1!$

Solution:
Option(D) is correct

\begin{align*}
1×1!\\
&= (2 -1)×1!\\
&= 2×1! - 1×1! \\
&= 2! - 1!\\
2×2!\\
&= (3 - 1)×2!\\
&= 3×2! - 2!\\
& = 3! - 2!\\
3×3!\\
&= (4 - 1)×3! \\
&= 4×3! - 3! \\
&= 4! - 3!\\
&\cdots\\
&\cdots\\
&\cdots\\
n×n! \\
&= (n+1 - 1)×n! \\
&= (n+1)(n!) - n! \\
&= (n+1)! - n!
\end{align*}

Summing up all these terms, we get $(n+1)! - 1!$

Edit: Thank you david for pointing out the error. Corrected the question.

## (5) Comment(s)

Saim
()

add and subtract 1 to each term. you get the answer within 30 secs :)

Sai
()

simply verify options

David
()

There is a mistake in the question...

it should be $3 \times 3!$ instead of $3! \times 3!$

Deepak
()

Thank you for letting me know the error. Corrected

PRATYUSH ANAND
()

Really Good+ Question