Problem 54 - Algebra for College Students

Question

How many different five-person committees can be formed from a group of 50 people? \(2,118,760\)

Step-by-Step Solution

Answer

2,118,760 committees can be formed.

Step 1: Understand the Problem

We need to determine how many different groups (committees) of 5 people can be selected from a total of 50 people. This is a combination problem because the order in which we select people does not matter.

Step 2: Identify the Formula

For problems involving combinations where order does not matter, we use the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 50 \) and \( r = 5 \).

Step 3: Substitute Values into the Formula

Insert the values into the formula: \( \binom{50}{5} = \frac{50!}{5!(50-5)!} \). Simplify to \( \frac{50!}{5! \cdot 45!} \).

Step 4: Simplify the Expression

To simplify, notice that \( 50! = 50 \times 49 \times 48 \times 47 \times 46 \times 45! \), which allows \( 45! \) in the numerator and denominator to cancel out. Thus, it becomes \( \frac{50 \times 49 \times 48 \times 47 \times 46}{5!} \).

Step 5: Calculate the Denominator

Calculate \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Step 6: Calculate the Numerator

Multiply the numbers: \( 50 \times 49 \times 48 \times 47 \times 46 = 254251200 \).

Step 7: Compute the Division

Divide the result from Step 6 by 120: \( 254251200 \div 120 = 2118760 \).

Step 8: Present the Result

The number of different five-person committees that can be formed is 2,118,760.

Key Concepts

Combination Formula

In mathematics, a combination refers to a way of selecting items from a larger set, where the order of selection does not matter. This presents a distinct scenario compared to permutations, where order does matter. The formula for finding combinations is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \( n \) represents the total number of elements to choose from, and \( r \) is the number of elements to select. The exclamation point "!" denotes a factorial, which is a key concept in this formula.

The combination formula is particularly useful in situations such as forming teams, choosing lottery numbers, and the problem discussed here: forming committees. Since the order doesn’t matter in committee formation, we use this formula to find the total number of possible groups.

Factorial Calculations

Factorials are a fundamental part of calculating combinations, as seen in the combination formula. A factorial of a number \( n \) (denoted as \( n! \)) is the product of all positive integers from 1 to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In the context of combinations, factorial calculations help to determine how many different ways we can arrange a subset of a set. Factorials grow very quickly: even small numbers can result in large factorials. This rapid growth is why they are powerful tools when dealing with large datasets.

Knowing how to simplify factorials by canceling out common terms is extremely useful. For instance, in the problem where we have to calculate \( \binom{50}{5} \), we can reduce \( \frac{50!}{45!} \) by cancelling out \( 45! \) in both the numerator and denominator, simplifying the computation to just products and divisions of smaller numbers.

Selection Problems

Selection problems are a category of problems in mathematics that involve choosing a subset of items from a larger set. They often require understanding whether the problem is asking for a combination or permutation. With combinations, like in the committee problem here, order does not matter, whereas with permutations, it does.

These problems often appear in real-world scenarios like selecting team members, lottery numbers, or survey respondents. For instance, the original exercise—a classic example of a selection problem—asks for the number of five-person committees that can be formed from a group of 50. Since the sequence of selection is irrelevant, it is a combination problem.

Key strategies in solving selection problems include:

• Identifying the total number of items (\( n \)).
• Determining how many items you are selecting (\( r \)).
• Using the combination formula \( \binom{n}{r} \) if order does not matter.
• Executing the factorial calculations needed.

Mastering these strategies not only aids in mathematical computation but also enhances problem-solving skills in various fields.