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Chapter 33: Problem 1069

Determine the number of permutations of the letters in the word BANANA.

Short Answer

Expert verified
There are 60 permutations of the letters in the word BANANA, using the formula for permutations with repetition: \( \dfrac{6!}{1!3!2!} = \dfrac{720}{12} = 60 \).

Step by step solution

01

Identify unique letters and their counts

First, we need to identify unique letters in the word BANANA and count their frequency. Here, we have three unique letters: B, A, and N. B - 1 time A - 3 times N - 2 times
02

Determine the total number of letters

Next, we need to determine the total number of letters in the word BANANA. There are 6 letters in total.
03

Apply the formula for permutations with repetition

In this step, we apply the formula for permutations with repetition, which is: \( \dfrac{N!}{n_1!n_2!...n_k!} \) Where N is the total number of elements in the set and \(n_1, n_2, ..., n_k\) are the number of repetitions of each element. In our case, N=6 (total number of letters in BANANA), \(n_1 = 1\) (number of times B appears), \(n_2 = 3\) (number of times A appears), and \(n_3 = 2\) (number of times N appears).
04

Calculate the permutations with repetition

Now, we apply the formula: \( \dfrac{6!}{1!3!2!} \) Calculate the factorials: \( \dfrac{720}{1*6*2} \) Divide: \( \dfrac{720}{12} = 60 \)
05

Determine the number of permutations

There are 60 permutations of the letters in the word BANANA.

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