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

Answer:

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Answer 2

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:

                        $P_{r} ^{n} = n!/(n-r)!$

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 / Arragments reduce due to duplicate letters

          $= (8!)/(2! * 2!)$

          $= (8*7*6*5*4*3*2*1)/(2*1 * 2*1)$

          $= (8*7*6*5*3*2*1)$

          $= 10080$

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

To learn more about Permutations, visit

https://brainly.in/question/36535787

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