Question

How many unique ways are there to arrange the letters of the word “PRIOR”.
(a) 60
(b) 120
(c) 180
(d) 150

Answer 1

Hint: To solve this question, we will first of calculating the number of letters in PRIOR. Finally, we will count as “R” occurs two times. So, we will divide by 2 the required arrangements obtained. Arrangements of n numbers are given by n!.

Complete step by step answer:
We are given the word PRIOR. The number of letters on the word PRIOR is given by 5. And in that word PRIOR, the letter is repeated twice. We have to calculate the unique ways to arrange the letters of the word “PRIOR”. We have the number of times each letter is given by
\[P=1\]
\[R=2\]
\[I=1\]
\[O=1\]
So, we have 4 different letters. The total number of arrangements of the words “PRIOR” is given by dividing all the arrangements of the number of letters, i.e. 5 by 2 as the letter “R” is repeated twice. The number of arrangements of n letter words is given by n!.
Here, n = 5, so the number of arrangements is given by 5!
Hence, the total number of unique arrangements of the letter PRIOR is
\[\Rightarrow \dfrac{5!}{2}\]
\[\Rightarrow \dfrac{5\times 4\times 3\times 2\times 1}{2}\]
\[\Rightarrow 60\]

Hence, there are 60 unique ways to arrange the letters of the word “PRIOR”.

Note: Here as we need the unique ways, so we have divided by 2, as “R” is repeated twice. If it was given that we do not have to compute the unique ways to do so then the answer would be given by n! that is here it is 5! And n! formula is \[n!=n\times \left( n-1 \right)\times \left( n-2 \right)\times .....\times 1.\]