In how many different ways can the letters of the word 'RUMOUR' be arranged?

180

The correct answer is Option (2) → 180

1. Analyze the word

First, count the total number of letters and identify those that repeat:

• Total number of letters ($n$): 6 (R, U, M, O, U, R)
• Repetitions:
• The letter 'R' appears 2 times.
• The letter 'U' appears 2 times.
• Letters 'M' and 'O' appear 1 time each.

2. The Formula

The number of distinct arrangements for $n$ objects where some are identical is:

$\text{Total Arrangements} = \frac{n!}{p! \times q! \times \dots}$

Where:

• $n!$ is the total number of letters.
• $p!, q!$, etc., are the factorials of the counts of each repeating letter.

3. Calculation

Substitute the values for 'RUMOUR':

$\text{Arrangements} = \frac{6!}{2! \times 2!}$

Calculate the factorials:

• $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
• $2! = 2 \times 1 = 2$

Now, complete the division:

$\text{Arrangements} = \frac{720}{2 \times 2}$

$\text{Arrangements} = \frac{720}{4}$

$\text{Arrangements} = 180$