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Chapter 11: Problem 63

If 24 permutations can be formed using the letters in the word BAKE, why can't 24 permutations also be formed using the letters in the word BABE? How is the number of permutations in BABE determined?

Short Answer

Expert verified
The word 'BABE' can only create 12 permutations rather than 24. This is due to the repetition of the letter 'B'. The formula used to calculate permutations must be adjusted when there is repetition, necessitating that the total number of characters factorial be divided by the factorial of the number of times the repeating character appears.

Step by step solution

01

Understand Permutations

Permutations refer to the arrangement of objects in a specific order. If we have \(n\) distinct objects, they can be arranged in \(n!\) ways, where \(n!\) is the factorial of \(n\) and represents the product of all positive integers up to \(n\).
02

Calculate Permutations for the Word BAKE

For the word 'BAKE', we have 4 distinct letters. We can calculate the total number of permutations using the formula \(n!\), where \(n\) is the total number of letters. Here \(n\) is 4, so the total number of permutations is \(4!\) which equals 24.
03

Reasoning for the Word BABE

In the word 'BABE', we have 4 letters, but one of them, 'B', repeats twice. The repetition of a character in a selection changes the method of finding the count of the permutation. Now, we need to divide the factorial of the total number of characters by the factorial of the number of times the repeating character is appearing. This is because when items are identical, it does not create a new arrangement when they are swapped.
04

Calculate Permutations for the Word BABE

For the word 'BABE', we have a total of 4 letters with 'B' repeating twice. We now apply the formula \(\frac{n!}{p!}\) where \(n\) is the total number of letters (4) and \(p\) is how many times 'B' repeats (2). We find that the total permutations equals \(\frac{4!}{2!} = 12\).

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