how many distinguishable permutations are there of the letters in the word effective

240

To find the number of distinguishable permutations of the letters in the word "effective," we can use the concept of permutations with repeated elements.

The word "effective" has 9 letters in total, including 3 'e's and 2 'f's. The remaining 4 letters ('c', 't', 'i', and 'v') are all distinct.

To calculate the number of distinguishable permutations, we need to divide the total number of permutations (which is the factorial of the total number of letters) by the factorials of the repeated letters.

The calculation for the number of distinguishable permutations is as follows:

Total number of letters: 9
Number of repeated 'e's: 3
Number of repeated 'f's: 2

Distinguishable permutations = 9! / (3! * 2!)

Let's calculate this:

9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880
3! = 3 * 2 * 1 = 6
2! = 2 * 1 = 2

Distinguishable permutations = 362,880 / (6 * 2) = 60,480

Therefore, there are 60,480 distinguishable permutations of the letters in the word "effective."