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Find the value of ${}^8{P_7},{}^{25}{P_5},{}^{24}{P_4},{}^{19}{P_{14}}.$

Last updated date: 19th Mar 2023
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Hint: In this question we need to find values of the given permutation values. We will use the formula to compute the permutation, that is ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$. This will help us to solve the expressions.

Complete step-by-step answer:
For ${}^8{P_7}$, to solve this we will use the formula ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$.
Using the formula, we get ${}^8{P_7} = \dfrac{{8!}}{{\left( {8 - 7} \right)!}} = 8!$
$ \Rightarrow {}^8{P_7} = 40320$
Similarly, for \[{}^{25}{P_5}\] we get,
${}^{25}{P_5} = \dfrac{{25!}}{{\left( {25 - 5} \right)!}} = \dfrac{{25!}}{{20!}} = 25 \times 24 \times 23 \times 22 \times 21$
$ \Rightarrow {}^{25}{P_5} = 6375600$
Similarly, for \[{}^{24}{P_4}\] we get,
${}^{24}{P_4} = \dfrac{{24!}}{{\left( {24 - 4} \right)!}} = \dfrac{{24!}}{{20!}} = 24 \times 23 \times 22 \times 21$
$ \Rightarrow {}^{24}{P_4} = 255024$
Similarly, for \[{}^{19}{P_{14}}\] we get,
${}^{19}{P_{14}} = \dfrac{{19!}}{{\left( {19 - 14} \right)!}} = \dfrac{{19!}}{{5!}}$
$ \Rightarrow {}^{19}{P_{14}} = = \dfrac{{19!}}{{5!}}$

Note: Whenever we face such types of problems the value point to remember is that we need to have a good grasp over permutations and its formulas. The most basic formula to calculate permutations has been discussed above and used to solve the given question. However, we must remember that we don’t need to calculate the value of factorial if it is very large.