Question

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

Answer 1

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.