Question

How many different permutation can be made using all letter in the word social science ?

Answer 1

`(13!)/(2!3!2!2!)=("6,227,020,800")/(2xx6xx2xx2)="129,729,600"`

Explanation:

The words "social science" contain 13 letters in all, and of those, we have 2 Ss, 3 Cs, 2 Is, and 2 Es.

If all 13 letters were unique, we'd have `13! = "6,227,020,800"` ways to arrange the letters.

However, because we have duplicate letters, we have to divide out the duplicate arrangements (for instance, note that `S_1S_2...` is the same as `S_2S_1...`). The number of ways the duplicate letters can be internally arranged is equal to how many there are, factorial (and so for the Ss it's `2! =2`, for the Cs its `3! =6`, etc). This gives:

`(13!)/(2!3!2!2!)=("6,227,020,800")/(2xx6xx2xx2)="129,729,600"`