Question

The letters in the word HORSES are arranged in a row.How many unique arrangements are there of the letters?

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_____ arrangements

Answer 1

180

Step-by-step explanation:

The word HORSES has 6 distinct letters, so we can arrange them in 6! = 720 ways if all the letters were unique.

However, in this case, the letters E and S are repeated twice. Therefore, we must divide by the number of arrangements of the repeated letters to avoid counting the same arrangement more than once.

The letter E appears twice, so there are 2! = 2 ways to arrange the E's within the word. Similarly, the letter S appears twice, so there are 2! = 2 ways to arrange the S's within the word.

Thus, the total number of unique arrangements of the letters in the word HORSES is:

6! / (2! × 2!) = 720 / 4 = 180

Therefore, there are 180 unique arrangements of the letters in the word HORSES.

Answer 2

There are 720 unique arrangements of the letters in the word HORSES.

Step-by-step explanation:

To find the number of unique arrangements of the letters in the word HORSES, we can use the concept of factorial. The word HORSES has 6 letters, so there are 6 choices for the first letter, 5 choices for the second letter, 4 choices for the third letter, and so on. Therefore, the number of unique arrangements is 6! (6 factorial), which is equal to 6 x 5 x 4 x 3 x 2 x 1 = 720 arrangements.