Problem 24 - Algebra and Trigonometry

Question

These problems involve permutations. Contest Prizes In how many different ways can first, second, and third prizes be awarded in a game with eight contestants?

Step-by-Step Solution

Answer

There are 336 ways to award the prizes to three contestants out of eight.

Step 1: Understanding Permutations

First, recognize that this is a permutation problem. We are looking to arrange 3 items (positions: first, second, third) out of 8 contestants. Permutations are used when the order is important.

Step 2: Setting Up the Permutation Formula

Use the permutation formula since the order of selection matters. The permutation of selecting r items from n is given by the formula: \( P(n, r) = \frac{n!}{(n-r)!} \), where \(!\) denotes factorial.

Step 3: Applying the Numbers to the Formula

Apply the numbers to the permutation formula where \( n = 8 \) and \( r = 3 \). Thus, we have the formula: \( P(8, 3) = \frac{8!}{(8-3)!} \). Calculate \( 8! \) and \( 5! \).

Step 4: Calculating Factorials

Calculate the values of 8 factorial and 5 factorial: \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] and \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 \]. Simplify to find \( 8! = 40320 \) and \( 5! = 120 \).

Step 5: Simplifying the Permutation Calculation

Substitute back into the permutation formula: \[ P(8, 3) = \frac{8!}{5!} = \frac{40320}{120} = 336 \]. Thus, there are 336 different ways to award the prizes.

Key Concepts

Factorial

A factorial, denoted as \(n!\), is a mathematical operation applied to a positive integer to find the product of all positive integers less than or equal to the number. For example, 5 factorial (written as 5!) is calculated as follows:

• 5! = 5 × 4 × 3 × 2 × 1 = 120.

Factorials grow very quickly as the number increases.
They play a critical role in permutation calculations to determine the total number of possible arrangements.
To illustrate, in the case of 8 contestants, 8! equals 40,320. That represents all the possible orders those 8 can be arranged.
For multiple factorial calculations like 5! from the total, it’s crucial to remember it simplifies calculations by breaking down the permutations further.

Permutation Formula

When considering permutations, it’s imperative to realize that the order of arrangement is essential.
The permutation formula is used to determine how many different ways you can arrange a number of items, out of a larger pool, when order matters.
For selecting r items from a larger group n, use the formula:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

Here, \(n\) refers to the total items to choose from, \(r\) signifies the number you are selecting.
In our example, 8 overall contestants with 3 prize positions means using \(P(8, 3)\).
By calculating 8!, you find all possible contestant arrangements. By dividing by 5!, you subtract out unneeded arrangements, resulting in 336 unique ways to distribute awards.

Order of Selection

Order of selection becomes vital when determining rankings, like awarding prizes.
In permutations, not only what's selected but their position or sequence matters.
For instance, a different position in a ranked setup creates a new permutation.
Since awarding the first prize to one individual versus another in a lineup changes the outcome, each contestant's position matters.

• This is unlike in combinations, where the positions hold no significance.
• In the exercise, the sequence of first, second, and third places among 8 contestants generated distinct results due to their unique ranking.

Understanding the order of selection and its impact on permutation calculations is key to solving problems where arrangement order is prioritized.