Problem 65 - Algebra and Trigonometry

Question

Solve the problem using the appropriate counting principle(s). Choosing a Committee A committee of six is to be chosen from a class of 20 students. The committee is to consist of a president, a vice president, and four other members. In how many different ways can the committee be picked?

Step-by-Step Solution

Answer

There are 1,162,800 ways to form the committee.

Step 1: Select the President

Firstly, we choose the president from the 20 students. There are 20 options available. This step can be represented as choosing 1 student out of 20, which gives us 20 possible choices.

Step 2: Select the Vice President

After selecting the president, we need to choose the vice president from the remaining students. Now, there are 19 students left. Thus, we have 19 options to choose the vice president.

Step 3: Select the Other Four Members

With the president and vice president selected, there are 18 students remaining. We need to select the remaining four members from these students. This requires a combination calculation, as the order does not matter. We represent this as \( \binom{18}{4} \), the number of ways to choose 4 members from 18.

Step 4: Calculate Combinations

Calculate \( \binom{18}{4} \) using the combination formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Substituting into this formula gives \( \binom{18}{4} = \frac{18 \times 17 \times 16 \times 15}{4 \times 3 \times 2 \times 1} = 3060 \).

Step 5: Multiply the Results

To find the total number of ways to form the committee, multiply the results from each step: 20 choices for president, 19 choices for vice president, and 3060 ways to choose the other four members. The total number of ways is 20 × 19 × 3060.

Step 6: Final Calculation

Calculate the multiplication: 20 × 19 × 3060 = 1162800. Thus, there are 1162800 different ways to form this committee.

Key Concepts

Permutation

A permutation is an arrangement of objects in a specific order. It's all about how many "arrangements" you can make with a set group. Every time you reorder the arranged objects, it counts as a different permutation.
To understand permutations, consider that the order in which you select members for a committee or a team matters.

• If you choose different individuals for specific roles, like president and vice president, that's an example of a permutation.
• For example, choosing a president from a class of 20 students involves 20 options, while the vice president is chosen from the remaining 19 students, leading to different permutations of the committee leadership.

The formula for a permutation where you choose "r" objects from "n" is given by:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
This differs from combinations, where order doesn't matter.

Combination

Combinations refer to selecting items from a group where order does not matter. In the context of choosing committee members, combinations are used when we are interested in how many ways we can choose a subset of members without caring about the order in which they are selected.
When we calculate a combination, we're simply figuring out how many different groups of items can be selected together, regardless of their sequence.
An example often illustrated is in selecting four regular members from a remaining group of 18 students once a president and vice president have been chosen. This calculation uses:

• The combination formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• Applying it, we find there are \( \binom{18}{4} = 3060 \) ways to choose these four members.

Combinations distinguish themselves from permutations by focusing solely on the groupings rather than the order of arrangement.

Factorial

Factorials are a mathematical operation represented by an exclamation mark \(!\). It signifies the product of all positive integers up to a specified number.
For example, \(5!\) means \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
In the contexts of permutations and combinations, factorials are fundamental. They help determine the number of ways a set can be arranged or grouped:

• In permutations, factorials account for arranging items in specific orders.
• In combinations, they help model the number of ways to choose subsets, as in the step \( \binom{18}{4} = \frac{18!}{4!(18-4)!} \), where factorials are pivotal in dividing and simplifying outcomes.

Understanding factorials is key to solving problems in combinatorial mathematics, as they form the basis of more complex calculations in these topics.