Question

A code is to be made by arranging 7 letters. Three of the letters used will be the letter \(A\), two of the letters used will be the letter \(B\), one of the letters used will be the letter \(C\), and one of the letters used will be the letter \(D\). If there is only one way to present each letter, how many different codes are possible? a. 42 b. 210 c. 420 d. 840 e. 5,040

Short answer

420 codes are possible.

Step-by-step solution

- Understand the Problem

We need to find the number of different ways to arrange 7 letters where the frequency of letters is predetermined: three A's, two B's, one C, and one D.

- Determine the Formula

We will use the permutation formula for non-distinct items: \[\frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdot n_4!}\] where \(n\) is the total number of items, and \(n_1, n_2,...\) are the frequencies of the distinct items.

- Assign the Values

Here, \(n = 7\) (total letters), \(n_1 = 3\) (three A's), \(n_2 = 2\) (two B's), \(n_3 = 1\) (one C), \(n_4 = 1\) (one D).

- Plug Values into the Formula

Substitute the values into the formula: \[\frac{7!}{3! \cdot 2! \cdot 1! \cdot 1!}\]

- Calculate Factorials

\(7! = 5040\), \(3! = 6\), \(2! = 2\), \(1! = 1\). Plugging these into the formula gives: \[\frac{5040}{6 \cdot 2 \cdot 1 \cdot 1} = \frac{5040}{12} = 420\]

- Conclusion

The number of different codes possible is 420.

Key concepts

Factorials in Permutations

In mathematics, a factorial is a function that multiplies a number by all the positive integers below it. It's denoted by an exclamation mark (!). For example, the factorial of 4 (written as 4!) is calculated as: 4! = 4 × 3 × 2 × 1 = 24.
Factorials are crucial in permutations as they help calculate the number of ways to arrange a set of items. In our exercise, we used factorials to determine both the total number of arrangements (7!) and the individual frequencies (3!, 2!, 1!, 1!). These helped us find the correct number of distinct permutations given the letter frequencies.

Permutation Formula

Permutations refer to the different ways of arranging a set of items. When items are repeated, we use a specific formula to calculate permutations. The permutation formula for non-distinct items is: \ \[ \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdot n_4!} \]
Here, \( n \) represents the total number of items, and \( n_1 \), \( n_2 \), etc., represent the frequencies of each distinct item. This formula adjusts for the repeated items by dividing the total permutations \( n! \) by the factorials of the frequencies. In our example exercise, \( n = 7 \), \( n_1 = 3 \), \( n_2 = 2 \), \( n_3 = 1 \), and \( n_4 = 1 \). This resulted in: \ \[ \frac{7!}{3! \cdot 2! \cdot 1! \cdot 1!} \]

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations. It focuses on counting, arrangement, and combination of elements within sets. This field is essential when solving problems like our letter arrangement exercise.
In combinatorics, we consider how to count and arrange elements under given constraints. For instance, arranging 7 letters with specific frequencies involves applying combinatorics principles to find the number of distinct codes. Utilizing concepts such as factorials and the permutation formula, we can solve these kinds of problems systematically and accurately.

Arranging Letters

Arranging letters involves determining how many different ways we can organize a set of letters to form distinct sequences. When letters repeat, the calculations change, as repeated items reduce the number of unique arrangements.
In our example exercise, we arranged three A's, two B's, one C, and one D, resulting in a specific permutation problem. By using the permutation formula for non-distinct items, we accounted for the repetitions and calculated the total number of unique sequences. The final result showed that there are 420 different ways to arrange these 7 letters.