Question

Find the value of \[n\] if \[{}^n{P_4} = 5\left( {{}^n{P_3}} \right)\]?

Answer 1

Hint:As permutation is an ordered combination in which number of n objects taken r at a time is determined by permutation and hence, here for the given permutation arrangement the value of \[n\] can be found by applying the formula as \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\].

Complete step by step answer:
Permutation is a technique that determines the number of possible arrangements in a set, hence we need to find the value of \[n\].Apply the formula to find the value of \[n\],
\[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\] …………………………………………1
Let us write the given equation
\[{}^n{P_4} = 5\left( {{}^n{P_3}} \right)\] …………………………………………2
Substituting the given values of \[n\]and \[r\] in equation 1 we get
\[\dfrac{{n!}}{{\left( {n - 4} \right)!}} = 5 \times \dfrac{{n!}}{{\left( {n - 3} \right)!}}\]
After simplifying we get
\[1 = \dfrac{5}{{n - 3}}\]
Now, multiplying the denominator term with the LHS of numerator we get
\[n - 3 = 5\]
\[\Rightarrow n = 5 + 3\]
\[\therefore n = 8\]

Therefore, the value of \[n\] is \[8\].

Additional Information:
To find the number of Permutations the formula is as follows
\[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]
A combination is a selection of all parts of a set of objects, without regard to the order in which they are selected. Hence in combination order is not important.To find the number of Combinations the formula is as follows,
\[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
When the objects are repeat the number of permutations of \[n\] different things, taken \[r\] at a time is \[{n^r}\] in which each may be repeated any number of times in each arrangement, where \[{n^r}\] denotes that number of ways in which \[r\] places can be filled by \[n\] different things when each thing can be repeated \[r\] times.

Note:In Permutation order is important whereas in combination order is not important. To find \[n\] value the objects can be arranged in \[n\left( {n - 1} \right)\left( {n - 2} \right)....\] ways, in which this is represented as \[n!\] and in the same way we can find the number of combinations asked.