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Chapter 7: Problem 374

Find the number of permutations of the seven letters of the word "algebra."

Short Answer

Expert verified
The number of permutations of the seven letters of the word "algebra" is 1260.

Step by step solution

01

Identify the number of distinct elements in the given word.

In the word "algebra," there are 7 letters, where 2 of them are "a"s, 2 are "l"s and the rest are distinct. So we have, \(n(a) = 2\) \(n(l) = 2\) \(n(g) = 1\) \(n(e) = 1\) \(n(r) = 1\)
02

Calculate the factorial for the total number of letters.

We have a total of 7 letters in the word "algebra." So we'll calculate the factorial, which will be the total number of permutations for these 7 letters without taking the repetitions into account. Factorial is represented using the exclamation mark and calculated as follows: \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)
03

Calculate the factorial for the number of repetitions.

As we identified in step 1, there are 2 "a"s and 2 "l"s. Now, we will calculate the factorial for these repetitions: For "a": \(2! = 2 \times 1 = 2\) For "l": \(2! = 2 \times 1 = 2\)
04

Apply the permutation formula for non-distinct elements.

The formula for permutations of non-distinct elements or elements with repetitions is as follows: \( \frac{Total\ factorial}{Product\ of\ factorials\ for\ all\ the\ repetitions} \) In our case, \( \frac{7!}{(2!)(2!)} = \frac{5040}{(2)(2)} = \frac{5040}{4} = 1260 \)
05

State the number of permutations in the word "algebra."

The number of permutations of the seven letters of the word "algebra" is 1260.

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