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Chapter 1: Problem 8

How many different letter arrangements can be made from the letters (a) Fluke? (b) Propose? (c) Mississippi? (d) Arrange?

Short Answer

Expert verified
The number of different letter arrangements for the given words are: (a) Fluke: 120 (b) Propose: 2520 (c) Mississippi: 34,650 (d) Arrange: 1260

Step by step solution

01

(a) Fluke

For the word "Fluke", all the letters are distinct. To find the number of letter arrangements, we simply need to find the permutations of the 5 different letters. This can be calculated using the formula for permutations: \(n! = n(n-1)(n-2)...\) where n is the total number of distinct elements. In this case, n = 5. The number of letter arrangements for the word "Fluke" is: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
02

(b) Propose

For the word "Propose", we have 7 letters with one repeating letter, i.e., "P". To find letter arrangements, we need to find permutations while considering the repeating letter. The permutations of a word of n elements while considering a repeating element is given by the formula: \(n! / (p_1! \times p_2! \times...)\), where \(p_i\) is the count of the ith repeating element. In this case, n = 7 and the count of repeating element "P" is 2. The number of letter arrangements for the word "Propose" is: \(7! / (2!) = 7 \times 6 \times 5 \times 4 \times 3 = 2520\)
03

(c) Mississippi

For the word "Mississippi", we have 11 letters with 3 repeating letters: S (4 times), I (4 times), and P (2 times). We will use the formula for permutations with multiple repeating elements. Here, n = 11, count of S is 4, count of I is 4, and count of P is 2. The number of letter arrangements for the word "Mississippi" is: \(11! / (4! \times 4! \times 2!) = 34,650\)
04

(d) Arrange

For the word "Arrange", we have 7 letters with 2 repeating letters: A (2 times) and R (2 times). We will use the formula for permutations with multiple repeating elements. Here, n = 7, count of A is 2, and count of R is 2. The number of letter arrangements for the word "Arrange" is: \(7! / (2! \times 2!) = 7 \times 6 \times 5 \times 4 \times 3 = 1260\) In conclusion, the number of different letter arrangements for the given words are: (a) Fluke: 120 (b) Propose: 2520 (c) Mississippi: 34,650 (d) Arrange: 1260

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