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Chapter 6: Problem 3

Determine the number of strings that can be formed by ordering the letters given. SALESPERSONS

Short Answer

Expert verified
The total number of strings that can be formed by ordering the letters of the word SALESPERSONS is \( 12! / (3! * 1! * 1! * 2! * 1! * 1! * 1! * 1!) \)

Step by step solution

01

Determine the total number of letters

First, count the total number of letters in the word SALESPERSONS. There are 12 letters.
02

Determine the count of each individual letter

Next, find out how many times each individual letter appears in the word SALESPERSONS: S appears 3 times, A appears 1 time, L appears 1 time, E appears 2 times, P appears 1 time, R appears 1 time, O appears 1 time, N appears 1 time.
03

Apply the formula

Apply the formula: \( n! / (n1! * n2! * ... * nk!) \) = \( 12! / (3! * 1! * 1! * 2! * 1! * 1! * 1! * 1!) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutations
Permutations are all about arranging items in a specific order. When we want to know how many ways we can arrange a group of objects, permutations come into play. For example, if you have three items, A, B, and C, the permutations would be ABC, ACB, BAC, BCA, CAB, and CBA. This means there are 6 permutations of the letters.

When dealing with letters, like in the word "SALESPERSONS," we have to consider the order of each letter. However, some letters repeat, so we use permutations when considering arrangements that distinguish these repetitions.
Factorial
Factorials are a key concept in permutations. The factorial of a number, denoted as \( n! \), means you multiply all whole numbers from \( n \) down to 1. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials help us calculate the total number of permutations for a set of distinct objects.

When considering permutations of 12 letters, as in "SALESPERSONS," you begin with the factorial of the total number of letters, \( 12! \). This calculation provides the basis for finding out how the letters can be arranged.
Multinomial Coefficient
Multinomial coefficients extend the idea of factorials and permutations. They handle situations where items are alike, which is common in words with repeating letters. In our example "SALESPERSONS," the letter S repeats 3 times and E repeats 2 times, which affects the permutations.

The formula to handle such cases is given by:
  • \( \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!} \)
where \( n \) is the total number of items, and \( n_1, n_2, \ldots, n_k \) are the counts of repeating groups of items. Here, \( 12!/(3! \times 2! \times 1!) \) accounts for the total permutations while considering the repeats, accurately giving the count.
Counting Principle
The counting principle is a basic but powerful tool in combinatorics. It says that if you have multiple steps to complete a process, the total number of ways to complete all steps is the product of the number of ways to complete each step individually.

This principle is often implicit in problems involving permutations where we systematically account for various aspects鈥攍ike the sequence of letters in a word and adjustments for any repetitions. By careful consideration of each step, such as arranging distinct letters or adjusting for duplicates, the counting principle ensures an accurate tally of possible combinations.

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Most popular questions from this chapter

Prove that among 35 students in a class, at least two have first names that start with the same letter.

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