Question

How do you calculate permutations of words?

Answer 1

Hint:In order to evaluate the above, count the number of letters in the word if all are distinct simply calculate and if some are repetitive then use $\dfrac{{n!}}{{{m_a}!{m_b}!...{m_z}!}}$to calculate the permutations.

Complete step by step solution:
There are two possible situations on which the answer of the above depends:

1.If all the letters in the word are distinct or non-repetitive.

To calculate the number of permutations of a word, simply evaluate $n!$ where n is the no of letters in the word.

For example: permutations of word “HELP”
Here number of letters in the word is equal to 4
Therefore, the number of permutations
$
= 4! \\
= 4 \times 3 \times 2 \times 1 \\
= 24 \\
$

2. .If some letters in the word are repetitive or the same.

To calculate the number of permutations of a word, evaluate
$\dfrac{{n!}}{{{m_a}!{m_b}!...{m_z}!}}$

Where n is the number of letters in the word and ${m_a},{m_b}...{m_z}$are the occurrences of the
repeated letters in the word.

For example: permutations of word “COOK”
Here number of letters in the word is equal to 4 and ‘O’ is repeating 2 times

Therefore, the number of permutations
$
= \dfrac{{4!}}{{2!}} \\
= \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1}} \\
= 12 \\
$
Formula:
$C(n,r)\, = {\,^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
$p(n,r)\, = {\,^n}{P_r} = \dfrac{{n!}}{{(n - r)!}}$

Additional Information:
1.Factorial: The continued product of first n natural numbers is called the “n factorial “ and denoted
by $n!$.

2.Permutation: Each of the arrangements which can be made by taking some or all of number of
things are called permutations.
If n and r are positive integers such that $1 \leqslant r \leqslant n$, then the number of all
permutations of n distinct or different things, taken r at one time is denoted by the symbol
$p(n,r)\,or{\,^n}{P_r}$.
$p(n,r)\, = {\,^n}{P_r} = \dfrac{{n!}}{{(n - r)!}}$

3.Combinations: Each of the different selections made by taking some or all of a number of objects
irrespective of their arrangement is called a combination.
The combinations number of n objects, taken r at one time is generally denoted by
$C(n,r)\,or{\,^n}{C_r}$

Thus, $C(n,r)\,or{\,^n}{C_r}$= Number of ways of selecting r objects from n objects.
$C(n,r)\, = {\,^n}{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$

Note:1. Factorials of proper fractions or negative integers are not defined. Factorial n defined only for whole numbers.
2.Meaning of Zero factorial is senseless to define it as the product of integers from 1 to zero. So, we
define it as $0! = 1$.