Distinguishable Ways to Arrange Letters

The question asks for the number of unique ways the letters in the word "CHANCE" can be arranged. This is a problem of permutations with repetitions.

Permutation Formula for Repetitions

When arranging items where some items are identical, we use the formula:

$ \text{Number of arrangements} = \frac{n!}{n_1! n_2! \dots n_k!} $

Where:

• $n$ is the total number of letters.
• $n_1, n_2, \dots, n_k$ are the frequencies of each distinct repeated letter.

Applying the Formula to "CHANCE"

The word "CHANCE" has 6 letters.

• Total letters, $n = 6$.
• The letters are C, H, A, N, C, E.
• The letter 'C' is repeated 2 times. So, $n_1 = 2$.
• All other letters (H, A, N, E) appear only once.

Using the formula:

$ \text{Number of arrangements} = \frac{6!}{2!} $

Calculation Steps

1. Calculate the factorial of the total number of letters: $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$.
2. Calculate the factorial of the frequency of the repeated letter: $2! = 2 \times 1 = 2$.
3. Divide the result from step 1 by the result from step 2: $ \frac{720}{2} = 360 $

Therefore, there are 360 distinguishable ways to arrange the letters of the word CHANCE.