Problem 27 - Algebra for College Students

Question

How many different seven-letter permutations can be formed from four identical H's and three identical T's? 35

Step-by-Step Solution

Answer

35 permutations can be formed.

Step 1: Identify the Type of Permutation

This problem involves permutations of a multiset, as the letters are not all distinct. We have 4 H's and 3 T's that must be arranged.

Step 2: Apply the Formula for Multiset Permutations

When dealing with permutations of a set that contains repeated elements, use the formula: \[ \frac{n!}{n_1! imes n_2! \times ext{...} \times n_k!} \]where \( n \) is the total number of items to arrange, and \( n_1, n_2, ext{...}, n_k \) are the frequencies of each distinct item. Here, \( n = 7 \), \( n_1 = 4 \) (for H's) and \( n_2 = 3 \) (for T's).

Step 3: Calculate the Factorials

First, calculate the factorials:\( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)\( 4! = 4 \times 3 \times 2 \times 1 = 24 \)\( 3! = 3 \times 2 \times 1 = 6 \)

Step 4: Plug the Values into the Formula

Substitute the calculated factorials into the multiset permutation formula:\[ \frac{7!}{4! imes 3!} = \frac{5040}{24 \times 6} \]

Step 5: Perform the Division

Calculate the division:\( 24 \times 6 = 144 \)\( \frac{5040}{144} = 35 \)

Step 6: Conclude the Calculation

The result of the division gives the total number of distinct permutations of the letters, which is 35.

Key Concepts

Factorials

Factorials are a fundamental concept that often appear in permutations and combinations. To understand them, imagine lining up a series of items. The factorial of a number, denoted by an exclamation mark (!), represents the total number of ways you can arrange these items. For example, the factorial of 4, written as 4!, is calculated by multiplying all positive integers up to 4:

• 4! = 4 x 3 x 2 x 1 = 24.

This means if you have 4 unique items, you can organize them in 24 different ways.
Factorials grow very quickly. Even a relatively small number like 7 results in a large value:

• 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040.

Factorials are crucial when determining permutations, especially when repeated items are involved, as they help normalize the count of identical arrangements.

Arranging Identical Items

Arranging identical items involves a concept known as permutations of a multiset. When items in a set are not unique, calculating permutations becomes more complex than simply using the factorial of the total number of items.
Instead, we must adjust for the repeated items. For instance, if you have seven letters consisting of four H's and three T's, you would use a special formula.
This formula is:

• \[ \frac{n!}{n_1! \times n_2!} \]

Where:

• \( n \) is the total number of items (here, 7)
• \( n_1 \) is the number of identical items of type 1 (4 H's)
• \( n_2 \) is the number of identical items of type 2 (3 T's).

This formula removes the extra counts from identical items, ensuring that each unique permutation is counted only once.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arranging, and analyzing arrangements of elements within a set.
This discipline tackles various problems, such as determining the number of possible outcomes in a situation, like arranging letters or selecting items.
By using principles from combinatorics, we solve problems like the one provided, which asks us to arrange letters H and T. It involves calculating the total number of different sequences available based on given constraints, such as identical items or specific positions.
Combinatorics often employs concepts such as:

• Permutations, which are arrangements where order matters.
• Combinations, which involve selections where order does not matter.

The solutions rely heavily on factorial calculations and understanding permutations or combinations, allowing us to explore and solve real-world problems efficiently.