In how many different ways can the letters of the word 'RUMOUR' be arranged? Option 1: 720 Option 2: 180 Option 3: 45 Option 4: 30

Answer: 180. Solution: The word 'RUMOUR' has 6 letters total. Counting the frequency of each letter: - R appears 2 times - U appears 2 times - M appears 1 time - O appears 1 time The number of different arrangements with repeated letters is given by: Number of arrangements = (n!)/(n_1! x n_2! x ...) where n is the total number of letters and n_1, n_2, ... are the frequencies of each repeating letter. Substituting the values: Number of arrangements = (6!)/(2! x 2!) = (6 x 5 x 4 x 3 x 2 x 1)/(2 x 2) = 720/4 = 180 Therefore, the letters of 'RUMOUR' can be arranged in 180 different ways.

Quantitative Reasoning

Permutation & Combination

Medium

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

720

180

✅ Correct Option: 2

The word 'RUMOUR' has 6 letters total.

Counting the frequency of each letter:

R appears 2 times
U appears 2 times
M appears 1 time
O appears 1 time

The number of different arrangements with repeated letters is given by:

Number of arrangements $= \dfrac{n!}{n_1! \times n_2! \times ...}$

where $n$ is the total number of letters and $n_1, n_2, ...$ are the frequencies of each repeating letter.

Substituting the values:

Number of arrangements $= \dfrac{6!}{2! \times 2!}$

$= \dfrac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 2}$

$= \dfrac{720}{4}$

$= 180$

Therefore, the letters of 'RUMOUR' can be arranged in 180 different ways.

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Read the information given below carefully and answer the question that follows:

(A) The different ways in which the alphabets of the word BAKERY can be arranged is 720

(B) The number of ways in which the alphabets of the word MACHINE can be arranged so that the vowels will occupy only the odd positions is 576

Choose the correct answer from the options given below:

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

Solution

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