Question

The number of permutations of word DEPENDENT is:

• A. \( 15120 \)
• B. \( 15125 \)
• C. \( 16632 \)
• D. None of the above

Hint:

When finding permutations of a word with repeated letters, divide the factorial of the total number of letters by the factorial of the count of each repeated letter.

Answer 1

The Correct Option is A

Step 1: Count the total number of letters in the word "DEPENDENT". The word has 9 letters: D, E, P, E, N, D, E, N, T.

Step 2: Count the frequency of each distinct letter. - D appears 2 times - E appears 3 times - P appears 1 time - N appears 2 times - T appears 1 time Total letters \( n = 9 \).

Step 3: Apply the formula for permutations with repetitions. The number of distinct permutations of \( n \) objects where there are \( n_1 \) identical objects of type 1, \( n_2 \) identical objects of type 2, ..., \( n_k \) identical objects of type k is given by: \[ \frac{n!}{n_1! n_2! \cdots n_k!} \] In this case, \( n = 9 \), and the counts of repeated letters are \( n_D = 2 \), \( n_E = 3 \), \( n_N = 2 \).

Step 4: Substitute the values into the formula. \[ \text{Number of permutations} = \frac{9!}{2! \times 3! \times 2!} \]

Step 5: Calculate the factorial values and the final result. \[ 9! = 362,880 \] \[ 2! = 2 \] \[ 3! = 6 \] \[ 2! = 2 \] \[ \text{Number of permutations} = \frac{362,880}{2 \times 6 \times 2} = \frac{362,880}{24} \] \[ \text{Number of permutations} = 15,120 \]

Step 6: Compare the result with the given options. The calculated number of permutations is 15,120, which matches option (A).