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Permutations & Combinations

Permutation relates to arranging certain elements or things in an order. Number of permutations of 'n' distinct elements where 'r' elements of these are selected is given by $^{\text{n}} \text{P} _{\text{r}}$ = $\dfrac{\text{n}!}{(\text{n} - \text{r})!}$

So, as the definition suggests, permutation includes selection of elements and their arrangement.

Let's take a couple of examples. To understand the logic behind this formula.

Example 1: In how many ways can 4 students be seated on 4 chairs?

Number of students who can sit on the first chair is 4, the second is 3, third is 2 and fourth is the one remaining.

Therefore, number of permutations = 4 $\times$ 3 $\times$ 2 $\times$ 1 = 24

The same can be represented as
Total number of students = n = 4
Number of students to be arranged = r = 4

Total number of permutations = $^{\text{n}} \text{P} _{\text{r}}$ = $^{4} \text{P} _{4}$ = $\dfrac{4!}{(4 – 4)!}$ = $\dfrac{4!}{0!}$ = $24$
Example 2: In how many ways can 5 students be seated on 2 chairs?

Number of students who can sit on the first chair is 5 and the second is 4. Therefore, 5 $\times$ 4 = 20

Using the formula,
Total number of students = n = 5
Number of students to be arranged = 2
Total number of permutations = $^{5} \text{P} _{2}$ = $\dfrac{5!}{(5 – 2)!}$ = $\dfrac{5!}{3!}$ = $\dfrac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1}$ = $5 \times 4$ = $20$
So, the $^{\text{n}} \text{P} _{\text{r}}$ formula for number of permutations, simply put, is total number of ways n elements can be arranged [ n! ] divided by the number of ways the unselected elements can be arranged [ (n $-$ r)! ].