Question

How do you evaluate$^7{P_4}$?

Answer 1

This question is related to the topic of permutation. In this question we need to find the value of permutation $^7{P_4}$. To solve this we need to know the definition of permutation and factorial of a number. Permutation is a way of arranging objects in which order matters.

Complete step by step answer:
Let us try to solve this question in which we are asked to find the value of $^7{P_4}$. Before solving this we need to understand the definition of factorial and definition of permutation, formula to calculate permutation.
Factorial of a number $n$ means product of all numbers less than $n$ till $1$. For example a factorial of $3$ is equal to$1 \times 2 \times 3 = 6$. Factorial of a number is denoted by the symbol ’$!$’. We have an exception in the definition of factorial. Factorial of $0$ is $1$.
Permutation: permutation is ordering of objects taken some or all at a time. In permutation order of objects matter.
Formula for finding the permutation is given by
$^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$ Where $0 \leqslant r \leqslant n$

Now, we have all the required prerequisites to find the value of $^7{P_4}$. So let’s find it.
We have $n = 7$ and $r = 4$. So putting this values in formula, we get
$\Rightarrow ^7{P_4} = \dfrac{{7!}}{{\left( {7 - 4} \right)!}} \\\ \Rightarrow \dfrac{{7!}}{{3!}} \\\$
Now value of $7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5040$
And value of $3! = 3 \cdot 2 \cdot 1 = 6$
Value of $^7{P_4}$ is
\Rightarrow ^7{P_4} = \dfrac{{7!}}{{3!}} \\\ \Rightarrow ^7{P_4} = \dfrac{{5040}}{6} = 840 \\\
Hence the value of $^7{P_4}$ is $840$.

Note: This type of question in which we are asked to evaluate permutation or combination are asked for short answer type questions. To solve them you have to just apply the formula. Permutation and combination are used in problems of probability.