Question

If ${}^{56}p_{r+6}:{}^{54}p_r+3=30800:1$, then the value of $r$ is:

• A. $40$
• B. $51$
• C. $41$
• D. 510
• E. 101

Answer 1

The correct option is C

$41$

Explanation for the correct option:

Step 1: Expand the given expression:

Given:${}^{56}p_{r+6}:{}^{54}p_{r+3}=30800:1$

$\begin{array}{rcl}{}^{56}p_{r+6}:{}^{54}p_{r+3}& =& 30800\\ \frac{56!}{(56-r-6)!}:\frac{54!}{\left(54-r-3\right)!}& =& 30800\end{array}$ $\left[\because {}^{n}p_r=\frac{n!}{(n-r)!}\right]$

Step 2: Simplify the obtained equation:

Using basic algebraic methods, simplify the equation.

$\begin{array}{rcl}\frac{56!}{(50-r)!}:\frac{54!}{\left(51-r\right)!}& =& 30800\\ \frac{56\times 55\times 54!}{(50-r)!}\times \frac{(51-r)(50-r)!}{54!}& =& 30800\\ 51-r& =& \frac{30800}{56\times 55}\\ 51-r& =& 10\\ r& =& 41\end{array}$

So the value of $r$ is obtained as $41$.

Hence, option (C) is the correct answer.