Question

How do you evaluate ${}^7{P_7}$?

Answer 1

This problem deals with permutations and combinations. But here a simple concept is used. Although this problem deals with permutations only. Here factorial of any number is the product of that number and all the numbers less than that number till 1.
$\Rightarrow n! = n(n - 1)(n - 2).......1$
The number of permutations of $n$ objects taken $r$ at a time is determined by the formula which is used:
$\Rightarrow {}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$

Complete step-by-step solution:
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word “permutations” also refers to the act or process of changing the linear order of an ordered set.
Given we are asked to evaluate the given permutation expression which is ${}^7{P_7}$:
$\Rightarrow {}^7{P_7} = \dfrac{{7!}}{{\left( {7 - 7} \right)!}}$
$\Rightarrow {}^7{P_7} = \dfrac{{7!}}{{0!}}$
We know that $0! = 1$,
So $\Rightarrow {}^7{P_7} = 7!$
$\therefore {}^7{P_7} = 5040$

The value of ${}^7{P_7}$ is $5040$.

Note: Here while solving such kind of problems if there is any word of $n$letters and a letter is repeating for $r$times in it, then it can be arranged in $\dfrac{{n!}}{{r!}}$ number of ways. If there are many letters repeating for a distinct number of times, such as a word of $n$letters and ${r_1}$ repeated items, ${r_2}$repeated items,…….${r_k}$repeated items, then it is arranged in $\dfrac{{n!}}{{{r_1}!{r_2}!......{r_k}!}}$ number of ways.
Please note that permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms, in quantum physics, for describing the states of particles, and in biology, for describing RNA sequences.