Problem 34 - College Algebra

Question

Use the formula for \(_{n} P_{r}\) to solve Exercises \(33-40\) A corporation has ten members on its board of directors. In how many different ways can it elect a president, vice-president, secretary, and treasurer?

Step-by-Step Solution

Answer

The corporation can fill the four positions in 5040 different ways.

Step 1: Identify Variables

First, identify the total number of items and the number of items to choose from. In this case, there are 10 members in total (n=10) and 4 positions to be filled (r=4).

Step 2: Apply the Permutation Formula

The formula for permutation is \(_{n} P_{r} = \frac{n!}{(n-r)!}\). Substitute the values of n and r into the formula. Calculate the factorial of n and (n-r).

Step 3: Calculate the Result

Once you have calculated the factorials, divide the factorial of n by the factorial of (n-r). The result will give the number of ways the corporation can elect a president, vice-president, secretary, and treasurer.

Key Concepts

Combination and Permutation Formulas

When dealing with selection and arrangement problems, understanding when to use combinations and permutations is crucial. Combinations are used when the order of selection does not matter, while permutations are used when the order does matter. In the board election problem, since specific positions are being filled (president, vice-president, secretary, and treasurer), the order in which members are chosen is important. This means we use permutations.

The permutation formula is expressed as:

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

Here, \( n \) represents the total number of items, and \( r \) represents the number of items to choose and arrange.

In this problem, we are arranging 4 positions out of 10 board members, so \( n = 10 \) and \( r = 4 \). By applying the formula, we calculate the number of ways to fill these positions with order being important.

Factorials

Factorials are mathematical expressions used to determine the number of ways to arrange a set of items. Notated by an exclamation point \( ! \), a factorial of a number \( n \) (written \( n! \)) is the product of all positive integers less than or equal to \( n \).

For example:

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

Factorials are central to both combination and permutation calculations. They ensure that we account for every possible arrangement of a group of items.

In the board election problem, we deal with factorials in calculating permutations. We compute \( 10! \), which is the total arrangements of 10 members, and \( 6! \) (since \( 10 - 4 = 6 \)), which gives the arrangements to subtract when less than 10 members are considered. So, \( _{10} P_{4} = \frac{10!}{6!} \).

Board Election Math Problem

The board election problem is a perfect example of a permutation problem. The challenge is to determine in how many ways a specific number of people can be elected to different positions when there are multiple choices.

The problem involves choosing 4 board positions from 10 possible members. Using the permutation formula \( _{n} P_{r} = \frac{n!}{(n-r)!} \), we first find the factorial of 10 (\( 10! \)) and 6 (\( (10-4)! \)).

After computing these values, you divide \( 10! \) by \( 6! \) to get the result. This process shows the importance of order, as each position requires a distinct member. Hence, it's different from choosing a committee, where order doesn't matter.

This method helps solve many everyday scenarios where the arrangement or sequence is crucial, such as assigning distinct roles or ordering items uniquely.