Final answer:
To find the number of arrangements of the word 'unusual', calculate the factorial of the total number of letters (7!) and divide by the product of the factorials of the frequencies of the repeated letters (2! for 'u' and 2! for 'n'). The result is 1,260 unique arrangements.
Step-by-step explanation:
The question asks about the number of arrangements of the seven letters in the word unusual. In mathematics, particularly in combinatorics, this is a problem of finding the permutations of a multiset. The word unusual has 7 letters with repetitions (2 'u's, 2 'n's, 1 's', 1 'a', 1 'l'). The formula for permutations of a multiset is:
N! / (n1! × n2! × ... × nk!)
Where N is the total number of letters, and n1, n2, ..., nk are the frequencies of each distinct letter. So:
- Calculate the factorial of the total number of letters: 7! = 7×6×5×4×3×2×1.
- Calculate the factorials of the frequencies of the repeated letters: 2! for 'u' and 2! for 'n'.
- Divide the total factorial by the product of the factorial of the frequencies: 7! / (2!×2!).
The calculation gives the number of unique arrangements for the word unusual.
First, we find 7! = 5,040. Then, for the two 2! calculations, since 2! = 2, it's 2×2 = 4. Finally, divide 5,040 by 4 to get 1,260 arrangements.