Question

If \[{}^{56}{{P}_{r+6}}:{}^{54}{{P}_{r+3}}=30800:1\], find r.

Answer 1

Hint: The expression is that of Permutation, which represents ordered matters. For number of permutation of n things taken r at a time = \[{}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}\]. Simplify the given expression with this formula and find the value of r.

Complete step-by-step answer:
Permutation of a set is an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its element. Permutation is also the linear order of an ordered set. Thus the number of permutation (ordered matters) of n things taken r at a time is given as,
\[{}^{n}{{P}_{r}}=P\left( n,r \right)\dfrac{n!}{\left( n-r \right)!}\]
Now, we have been given that,
\[{}^{56}{{P}_{r+6}}:{}^{54}{{P}_{r+3}}=30800:1…...(1) \]
Let us simplify it as per the formula of Permutation.
\[{}^{56}{{P}_{r+6}}=\dfrac{56!}{\left( 56-r-6 \right)!}=\dfrac{56!}{\left( 50-r \right)!}\]
Similarly, \[{}^{54}{{P}_{r+3}}=\dfrac{54!}{\left( 54-r-3 \right)!}=\dfrac{54!}{\left( 51-r \right)!}\]
Now let us substitute the formula of \[{}^{54}{{P}_{r+6}}\] and \[{}^{54}{{P}_{r+3}}\] in Equation (1)
\[{}^{56}{{P}_{r+6}}:{}^{54}{{P}_{r+3}}=30800:1\]
\[\dfrac{56!}{\left( 50-r \right)!}:\dfrac{54!}{\left( 51-r \right)!}=30800:1\], we can write the above as,

\[\dfrac{\dfrac{56!}{\left( 50-r \right)!}}{\dfrac{54!}{\left( 51-r \right)!}}=\dfrac{30800}{1}\]
\[\Rightarrow \dfrac{56!}{\left( 50-r \right)!}\times \dfrac{\left( 51-r \right)!}{54!}=30800\] - (2)
We can write \[56!=56\times 55\times 54!\]
Similarly we can write, \[\left( 51-r \right)!=\left( 51-r \right)\left( 50-r \right)!\]
Now put these values in (1) and simplify it
\[\begin{align}
  & \dfrac{56\times 55\times 54!}{\left( 50-r \right)!}\times \dfrac{\left( 51-r \right)\left( 50-r \right)!}{54!}=30800 \\
 & 56\times 55\times \left( 51-r \right)=30800 \\
 & \Rightarrow 51-r=\dfrac{30800}{56\times 55}=10 \\
 & \therefore 51-r=10 \\
 & r=51-10=41 \\
\end{align}\]
Thus we got the required value of r = 41.

Note: Don’t confuse the formula of permutation with the formula of combination. In combination, the order does not matter. Thus the number of combinations will have the formula, n things taken r at a time:
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\].We can also verify the answer by substituting value r=41 in the expression \[{}^{56}{{P}_{r+6}}:{}^{54}{{P}_{r+3}}=30800:1\] and check whether L.H.S=R.H.S or not.