Question

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

Answer 1

The letters of the word "BROCCOLI" can be arranged in 10,080 different ways.

Calculating Arrangements :

The two counting terms in mathematics are Permutations and Combinations

Combinations are used in finding the number of ways of selecting a set of items from the given set.

Permutations are used to find the number of arrangements after selection from a given set of items.

The formulation of arranging and selecting 'r' items from 'n' items is:

Here, we need to find the number of arrangements using the letters of the word "BROCCOLI". So, we need to use Permutation. Also, the arrangement involves no selection as all the letters are to be used, we have n = r.

The total letters in the word "BROCCOLI" = 8

The unique letters in the word "BROCCOLI" are: B, R, L, I i.e. (4)

The other letter repetitions are :

The letter 'O' appears twice i.e. 2

The letter 'C' appears twice i.e. 2

So, the number of different arrangements possible are :

= Total arrangements / Arrangements reduce due to duplicate letters

Hence, the letters of the word Broccoli can be arranged in 10,080 ways.

Answer 2

10080

Step-by-step explanation:

To find the number of different ways that the letters of the word "broccoli" can be arranged, we can use the concept of permutations. The word "broccoli" has 8 letters, including 2 "o"s, and 2 "c"s.

To calculate the number of arrangements, we need to consider the total number of letters and account for any repetitions. We can use the formula for permutations with repeated elements:

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

Where:

n is the total number of elements

n1, n2, ..., nk are the number of repetitions for each element

In this case, the word "broccoli" has a total of 8 letters, with 2 "o"s and 2 "c"s. Substituting these values into the formula, we get:

8! / (2! * 2!)

Calculating this expression:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320

2! = 2 * 1 = 2

(40,320) / (2 * 2) = 40,320 / 4 = 10,080

Therefore, there are 10,080 different ways that the letters of the word "broccoli" can be arranged.