Question

Question 2. COUNTING. How many unique ways can the letters of the word LUMBERJACKS be arranged?

How many unique ways can the letters of the word HIGHLIGHT be arranged?
How many unique ways can the letters COOKBOOK be rearranged?
What is 88! divided by 86! ?

Answer 1

The letters of the word LUMBERJACKS can be arranged in 39,916,800 unique ways.

The letters of the word HIGHLIGHT can be arranged in 15,120 unique ways.

The letters of the word COOKBOOK can be rearranged in 840 unique ways.

88! divided by 86! is (88*87*86!)/(86!) = 7656

Explanation:

A word has n letters.

The are m repeating letters, each of them repeating
`r_(0), r_(1), ..., r_(m)` times

So the number of distincts ways the letters can be arranged is:

`N_(A) = (n!)/(r_(1)! * r_(2)! * ... * r_(m))`

LUMBERJACKS:

11 letters, none repeating. So

`N_(A) = (11!)/(0!) = 11! = 39916800`

The letters of the word LUMBERJACKS can be arranged in 39,916,800 unique ways.

HIGHLIGHT

9 letters

H repeats 3 times

G repeats 2 times

I repeats 2 times

`N_(A) = (9!)/(3!2!2!) = 15120`

The letters of the word HIGHLIGHT can be arranged in 15,120 unique ways.

COOKBOOK:

8 letters

O repeats 4 times

K repeats 2 times

`N_(A) = (8!)/(4!2!) = 840`

The letters of the word COOKBOOK can be rearranged in 840 unique ways.

What is 88! divided by 86! ?

The factiorial of a number n can be writen as

`n! = n * (n-1)!`

Then

`88 = 88 * 87! = 88 * 87 * 86!`

Then

`(88!)/(86!) = (88*87*86!)/(86!) = 7656`

88! divided by 86! is (88*87*86!)/(86!) = 7656