Question

A classic counting problem is to determine the number of different ways that the letters of millennium can be arranged. Find that number. The number of different ways that the letters of millennium can be arranged is

Answer 1

The correct answer is that the number of different ways that the letters of the word "millennium" can be arranged is 226,800

Explanation:

1. Let's review the information provided to us to answer the question correctly:

Number of letters of the word "millennium" = 10

Letters repeated:

m = 2 times

i = 2 times

l = 2 times

n = 2 times

2. The number of different ways that the letters of millennium can be arranged is:

We will use the n! or factorial formula, this way:

10!/2! * 2! * 2! * 2!

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

3'628,800/2*2*2*2 = 3'628,800/16 = 226,800

The correct answer is that the number of different ways that the letters of the word "millennium" can be arranged is 226,800

Answer 2

The word millennium can be arranged in 25,200 different ways.

Solving word problems involving Permutations.

In combinatorics, permutations refers to different way in which an item can be arranged while taking the order of arrangement into consideration.

To find the permutation of the different ways that the letter Millennium is arranged, we need to know if any number is repeated or not. By doing so, we found out that:

m = 2 times,

i = 3 times

l = 2 times

e = 1 time

n = 1 time

u = 1 time

So, using the permutation formula, we have:

`P(10; 2,3,2,1,1,1) = (10!)/(2!* 3!* 2!* 1!* 1!* 1!)`

`P(10; 2,3,2,1,1,1) = (3628800)/(144)`

`P(10; 2,3,2,1,1,1) = 25200`

Therefore, we can conclude that the word millennium can be arranged in 25,200 different ways.