Question

How many 5 letter words , with or without meaning can be formed out of the letters of the word ‘EQUATIONS’ if repetition of letters is not allowed ?

Answer 1

Hint:
We can see that the number of letters in the word EQUATIONS is 9 and we asked to find the number of 5 letter words. Whenever we are given n letters and the number of r letter words formed with the n letters is given by ${}^n{P_r}$, and this can be solved using the formula ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$

Complete step by step solution:
We are given the word ‘EQUATIONS’
The number of letters in the word is 9
And we can see that all the nine letters are unique
Now we are asked to find the number of 5 letter words formed from these letters and repetition is not allowed
Whenever we are given n letters and the number of r letter words formed with the n letters is given by ${}^n{P_r}$
Here we know that n is 9 and r is 5
$ \Rightarrow {}^9{P_5}$
We know that ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
Using this we can get
$
   \Rightarrow {}^9{P_5} = \dfrac{{9!}}{{\left( {9 - 5} \right)!}} \\
   \Rightarrow {}^9{P_5} = \dfrac{{9!}}{{4!}} \\
   \Rightarrow {}^9{P_5} = \dfrac{{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{4 \times 3 \times 2 \times 1}} \\
   \Rightarrow {}^9{P_5} = 72 \times 42 \times 5 = 3024 \times 5 = 15120 \\
 $

Therefore , 15120 five letter words can be forms with the letters of the word EQUATIONS without repetition.

Note:
1) Same way if we are given a n letter word and asked to find the number of words which can be formed using those n letters can be given by $n!$
2) We don’t use the formula here as we are specified to find the number of r letter words from the given n letter word.