Problem 71 - Precalculus with Limits

Question

A six-member research committee at a local college is to be formed having one administrator, three faculty members, and two students.There are seven administrators, \( 12 \) faculty members,and \( 20 \) students in contention for the committee. How many six-member committees are possible?

Step-by-Step Solution

Answer

The total number of possible six-member committees is 290,600.

Step 1: Calculate the combinations of administrators

There is only one administrator required and there are seven to choose from. So the number of combination for this is '‘7 choose 1', which can be represented as \(C(7, 1)\). Using the formula for combinations, this is calculated as: \(\frac{7!}{1!(7-1)!} = 7\)

Step 2: Calculate the combinations of faculty members

Three faculty members are required and there are twelve to choose from. So the number of combinations for this is '12 choose 3', which can be represented as \(C(12, 3)\). Using the definitions of 12!, 3! and (12 - 3)! in the formula for combinations, this is calculated as: \(\frac{12!}{3!(12-3)!} = 220\)

Step 3: Calculate the combinations of students

Two students are required and there are twenty to choose from. So the number of combinations for this is '20 choose 2', which can be represented as \(C(20, 2)\). Using the definitions of 20!, 2! and (20 - 2)! in the formula for combinations, this is calculated as: \(\frac{20!}{2!(20-2)!} = 190\)

Step 4: Calculate the total combinations

The total number of possible committees can be found by multiplying the number of combinations for each type of member together: \(7 * 220 * 190 = 290600\)

Key Concepts

Factorials

Factorials, denoted by the symbol "!", are crucial in calculations involving permutations and combinations. They can seem daunting at first, but they're just a simple concept. The factorial of a number is the product of all positive integers less than or equal to that number. For example,

• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)

This pattern continues for larger numbers, providing a way to calculate complex arrangements easily. In committee formation problems, factorials help us determine the number of ways to choose a specific subset from a larger group.

Committee Formation

Forming committees involves selecting a specific number of members from larger groups comprising different roles or categories, such as administrators, faculty members, and students. Each role might have a differing number of members required, contributing unique factors to the total calculation.
To determine all possible committee combinations, calculate the number of ways to choose individuals from each role or category and then multiply these counts.

For example, in our scenario, we pick:

• 1 out of 7 administrators
• 3 out of 12 faculty members
• 2 out of 20 students

Each calculation uses a specific combination formula, making committee formation a practical illustration of permutations and combinations concepts.

Binomial Coefficient

The binomial coefficient, often written as \(C(n, r)\), represents the number of ways to choose \(r\) objects from a set of \(n\) objects without regard to order. This mathematical tool is key in solving combination problems effectively.
You can calculate the binomial coefficient using the formula:\[C(n, r) = \frac{n!}{r! \times (n-r)!}\]For example, when selecting 3 faculty members from a total of 12, you calculate the binomial coefficient as \(C(12, 3)\).

• The numerator, \(12!\), accounts for all possible arrangements.
• The denominator, \(3! \times (12-3)!\), adjusts for the order not mattering and groups not being overcounted.

Utilizing the binomial coefficient simplifies solving various types of combination problems, like selecting committee members based on distinct categories.

Permutations and Combinations

Permutations and combinations are foundational concepts in combinatorics, and although they are related, each serves a distinct purpose.

• **Permutations** are used when the order of selection matters, involving arrangements of objects in a specific sequence.
• **Combinations** are used when the order does not matter, such as when forming a committee from a larger pool of candidates.

When forming a committee, as seen in the example, combinations are crucial because the order of selection isn't important – only the selection itself matters.

Using combinations ensures you're counting only the selections and not the different orders those selections can be made in. This distinction is vital to accurately calculating how groups and committees can be formed.