Problem 34 - Algebra 2

Question

How many ways can five members of the 100-member United States Senate be chosen to serve on a committee?

Step-by-Step Solution

Answer

There are 752,875 ways to choose 5 senators out of 100.

Step 1: Understand the Problem

You need to determine the number of ways to select a subset of senators from a larger group. Specifically, you are asked to choose 5 senators out of a total 100, which means this is a combination problem.

Step 2: Identify the Formula

For problems involving choosing a subset from a larger set where order does not matter, you use the combination formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]where \( n \) is the total number of items to choose from and \( k \) is the number of items to choose.

Step 3: Plug in the Numbers

Substitute the values into the formula. Here, \( n = 100 \) and \( k = 5 \). Thus, we have: \[ C(100, 5) = \frac{100!}{5!(100-5)!} \]which simplifies to \[ C(100, 5) = \frac{100!}{5! \times 95!} \]

Step 4: Calculate the Factorials

First, consider the factorials: - \( 100! = 100 \times 99 \times 98 \times 97 \times 96 \times 95! \) - \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)Thus, the expression becomes: \[ C(100, 5) = \frac{100 \times 99 \times 98 \times 97 \times 96}{5 \times 4 \times 3 \times 2 \times 1} \]

Step 5: Simplify the Expression

Calculate the numerator and denominator:- Numerator: \( 100 \times 99 \times 98 \times 97 \times 96 = 9034502400 \)- Denominator: \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \)Divide the numerator by the denominator: \[ \frac{9034502400}{120} = 752875 \]

Step 6: Verify the Answer

Double-check the calculations to ensure they have been performed correctly. The result, which represents the number of ways to select 5 senators from 100, should be consistent with standard calculations.

Key Concepts

Factorials

A factorial is a mathematical concept that involves multiplying a series of descending natural numbers. It's a way to express and calculate large products, especially useful in combinations and permutations. The symbol for factorial is the exclamation point

• For example, the factorial of 5 (written as 5!) is calculated as: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Factorials grow quite rapidly, with every increase in number resulting in much larger products. For instance, 6! would be \( 720 \).
• By definition, the factorial of zero is 1 (0! = 1), which might seem peculiar but maintains mathematical consistency.

Factorials are crucial in calculating combinations where order does not matter. For the problem of selecting members to form a committee, factorials simplify the process by considering different possible arrangements of the senate.

Committee Selection

When forming a committee from a larger group, the arrangement is usually irrelevant, and you only care about who is on the committee itself. This is an example of a combination problem.

• The committee selection is typical in scenarios like forming teams, allocating tasks, or planning group meetings.
• The task involves determining which members will be part of the group without assigning different roles or positions, making it distinct from permutations where order matters.

In committee selection problems, it's important to recognize that only the total number of members matters, not the specific arrangement in which they are chosen. This simplifies the calculation as you're only concerned with sets rather than sequences.

Combination Formula

The combination formula is used to determine the number of ways to choose a subset from a larger set where order does not matter. This principle is exemplified by the formula:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]Here,

• \( n \) is the total number of objects (like senators), and \( k \) is the subset size you want to choose (like committee members).
• The formula involves using factorials to calculate both the total number of arrangements (\( n! \)) and removing the duplicates by dividing through \( k! \), the factorial of the subset size, and \((n-k)!\), where \( n-k \) represents the elements not chosen.

This formula reflects the essence of combination problems, allowing you to calculate the number of unique groups rather than focusing on order. For example, in our exercise, you use the combination formula to find how 5 senators can be chosen from 100 total.

N Choose K

The term \( n \choose k \) (read as "n choose k") is shorthand for the combination formula. It's a concise way to express finding the number of combinations of \( n \) items taken \( k \) at a time.

• It is a commonly used notation in mathematics to describe combinations.
• In contexts such as the committee selection example, \( 100 \choose 5 \) is used to calculate how 5 senators are selected from the entire senate.
• It's a useful way to quickly refer to the combination calculation, and it encompasses more than just mathematics, as it can be applied to many real-world scenarios.

Understanding \( n \choose k \) helps simplify problems by providing a clear method to solve combination puzzles, allowing students to focus on applying formulas correctly rather than dwelling on theoretical complexities.