Question

The value of $\sum _{r=1}^{10}r{.}^r{P}_r$ is:

• A. ${}^{11}{P}_{11}$
• B. ${}^{11}{P}_{11}-1$
• C. ${}^{11}{P}_{11}+1$
• D. none of these

Answer 1

The correct option is B ${}^{11}{P}_{11}-1$

${\sum }_1^{10}{r}_r^r\text{P}$
$=1.1!+2.2!+3.3!+4.4!+5.5!+............+10.10!$
Adding and subtracting $1.1!$, we get
$=(1+1).1!+2.2!+3.3!+4.4!+5.5!+............+10.10!-1.1!$
$=2.1!+2.2!+3.3!+4.4!+5.5!+............+10.10!-1.1!$
$=(1+2).2!+3.3!+4.4!+5.5!+............+10.10!-1.1!$
Similarly repeating the previous steps, we get
$=11!-1$
Hence, option 'B' is correct.