To find the number of ways the letters of the word "rumour" can be arranged, we need to consider that some letters are repeated.
Step 1: Identify the letters and their frequencies
- r: 2 times
- u: 2 times
- m: 1 time
- o: 1 time
The word "rumour" has 6 letters in total.
Step 2: Use the formula for permutations of multiset
The formula for the number of distinct permutations of n letters where there are duplicates is:
n!n1!×n2!×⋯\frac{n!}{n_1!\times n_2!\times \cdots}n1!×n2!×⋯n!
where nnn is the total number of letters, and n1,n2,…n_1,n_2,\ldots n1,n2,… are the frequencies of the repeated letters. Here:
n=6,nr=2,nu=2n=6,\quad n_r=2,\quad n_u=2n=6,nr=2,nu=2
Step 3: Calculate the number of arrangements
6!2!×2!=7202×2=7204=180\frac{6!}{2!\times 2!}=\frac{720}{2\times 2}=\frac{720}{4}=1802!×2!6!=2×2720=4720=180
Final answer:
The letters of the word "rumour" can be arranged in 180 distinct ways.
