how many distinctly different arrangements are possible using every letter in the word:

intelligence

There are two i, three e, two l, two n.

C(12;12,2,3,2,2)

= 12!/(2!3!2!2!) and you know can reduce that down some. Check my thinking with your text material.

thanks

To find the number of distinctly different arrangements possible using every letter in the word "intelligence," we can use the concept of permutations.

The word "intelligence" has a total of 12 letters in it, with some letters repeating. In this case, we will use the permutation formula:

nPr = n! / (n1! * n2! * ... * nk!)

Where n is the total number of objects (in this case, letters), and n1, n2, ..., nk represents the number of repetitions for each letter.

In "intelligence," there are a total of 12 letters, but some letters repeat. Let's break it down:

- There are three 'e's.
- There are two 'i's.
- There are two 'n's.
- There is only one of each of the other letters.

Substituting these values into the permutation formula, we get:

12! / (3! * 2! * 2!)

Simplifying this expression:

12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
3! = 3 * 2 * 1
2! = 2 * 1

Now we can substitute these values:

(12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1) * (2 * 1))

Calculating this expression:

(479,001,600) / (6 * 2 * 2)

(479,001,600) / (24)

Resulting in: 19,958,400

Therefore, using every letter in the word "intelligence," there are 19,958,400 distinctly different arrangements possible.