Question

What is $^6{P_3}$?

Answer 1

In mathematics, a permutation of a set is a loose arrangement of its members into a sequence or linear order, or a rearrangement of its elements if the set is already ordered. The act or method of changing the linear order of an ordered set is often referred to as “Permutation”.

Complete step by step solution:
A permutation is a collection of objects arranged in a specific order. The members or elements of sets are arranged in this diagram in a linear or sequential order. For instance, the permutation of set A $= (12,8)$ is $2$, as in $12,8$ and $8,12$.
Formula of permutation:
The following is the formula for permutation of n objects for r selection of objects:
$P(n,r) = \dfrac{{n!}}{{(n - r)!}}$
Let’s find the permutation of $^6{P_3}$. Here we can observe that the n (no. of objects) is 6 and r is 3 (no. of objects selected).
Now, applying the above formula to find the permutation –
$P(6,3) = \dfrac{{6!}}{{(6 - 3)!}}$
Now, we have to expand the factorial. Firstly $6!$ and $(6 - 3)!$ can be written as –
$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
$(6 - 3)! = 3! = 3 \times 2 \times 1$
Now,
$P(6,3) = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{3 \times 2 \times 1}}$
$P(6,3) = 120$
The values of $^6{P_3}$ is 120.
So, we can say that there are $120$ ways of selecting $3$ objects out $6$ objects.

Additional Information:
Permutation can be categorized into three categories:

When $n$ different objects are there (when repetition is not allowed)
When repetition is permissible.
Where the points are not distinct, permutation is used (Permutation of multi sets)

Note:
Permutations are used in almost every branch of mathematics, as well as a wide range of other scientific disciplines. They're used to analyze sorting algorithms in computer science, describe particle states in quantum physics, and describe RNA sequences in biology.