how many distinguishable permutations are in the word MUSKETEERS?
http://answers.yahoo.com/question/index?qid=20070612200923AA4JBsH&show=7
I will assume that the 3 E's and the 2 S's are indistinguishable
so the number of ways would be 10!/(2!3!) = 302400
To find the number of distinguishable permutations in the word "MUSKETEERS," we can use the concept of permutation.
A permutation is an arrangement of objects in a specific order. In this case, we need to find the number of arrangements of the letters in the word "MUSKETEERS" where the arrangement is considered unique even if some letters are repeated.
To calculate the number of distinguishable permutations, we will use the formula for permutations with repeated elements. Let's break down the process:
1. Count the number of times each letter appears in the word:
- M appears once.
- U appears once.
- S appears once.
- K appears once.
- E appears three times.
- T appears once.
- R appears once.
2. Calculate the total number of permutations using the formula for permutations with repeated elements:
n! / (n1! × n2! × n3! ...)
where n is the total number of objects and n1, n2, n3, ... represent the repetitions of each object.
In this case, we have a total of 10 letters in "MUSKETEERS," with 1 repetition each for M, U, S, K, T, and R, and 3 repetitions for E.
So, the total number of distinguishable permutations is:
10! / (1! × 1! × 1! × 1! × 3! × 1! × 1!)
Now, let's calculate it:
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
1! × 1! × 1! × 1! × 3! × 1! × 1! = 6
Therefore, the number of distinguishable permutations in the word "MUSKETEERS" is:
3,628,800 / 6 = 604,800
So, there are 604,800 distinguishable permutations in the word "MUSKETEERS."