Problem 20 - Algebra and Trigonometry

Question

19–32 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.

Step 1: Understanding the Problem

We need to determine the number of ways to award first, second, and third prizes. This is a permutations problem because the order of awarding prizes matters.

Step 2: Identify Permutation Formula

To solve this problem, we use the permutation formula, which is given by \(P(n, r) = \frac{n!}{(n-r)!}\), where \(n\) is the total number of contestants, and \(r\) is the number of positions to fill (first, second, and third).

Step 3: Apply Values to the Formula

Here, \(n = 8\) (total contestants) and \(r = 3\) (positions for first, second, and third prizes). Substitute these into the formula: \[ P(8, 3) = \frac{8!}{(8-3)!} \] \[ P(8, 3) = \frac{8 \times 7 \times 6 \times 5!}{5!} \] The \(5!\) cancels out, leaving: \[ 8 \times 7 \times 6 \]

Step 4: Calculate the Permutation

Now, calculate the product: \(8 \times 7 \times 6 = 336\). This is the total number of ways to award the first, second, and third prizes.

Key Concepts

Permutation Formula

The permutation formula is essential in determining how many different sequences can be made from a set of items where the order matters. Think of it like organizing books on a shelf; rearranging them yields a different order. The mathematical formula for permutations is given by \[P(n, r) = \frac{n!}{(n-r)!}\]Here, \( n \) represents the total number of items available, while \( r \) signifies the number of positions we are arranging. The exclamation mark \(!\) denotes a factorial, which means it multiplies the number by all positive integers down to 1.
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This formula helps us calculate all potential arrangements where each sequence is distinct due to its order. Understanding and applying this formula correctly simplifies solving many arrangement problems.

Permutation Calculation

Once you understand the formula, making a permutation calculation is straightforward. Let's walk through a typical application:
Suppose you have eight contestants and need to award first, second, and third prizes. You need to calculate \( P(8, 3) \).
Substitute the values into the formula:

• \( n = 8 \)
• \( r = 3 \)

This becomes:\[P(8, 3) = \frac{8!}{(8-3)!} = \frac{8!}{5!}\]To simplify, calculate the factorials only as needed:

• \( 8! = 8 \times 7 \times 6 \times 5! \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

Cancel \( 5! \) in the denominator and numerator:\[8 \times 7 \times 6 = 336\]This calculation yields the number of different ways to award the prizes, stressing the importance of sequence.

Contest Prize Arrangement

When arranging contest prizes, thinking about all the different ways to award them is where permutations shine. This problem becomes engaging when you consider that each prize tier must have a unique participant. If we're assigning first, second, and third places:

• First place can potentially go to any of the 8 contestants.
• With one person taking the first spot, only 7 contestants remain for second place.
• After choosing second place, 6 contestants remain for third.

Thus, to count every possible arrangement using our permutation results, it illustrates howorder is crucial.
The success in calculations like \( 8 \times 7 \times 6 \) reveals that 336, as an outcome, emphasizes individual roles in the competitive setup. When you're preparing to solve these problems, focusing on the order and arranging freely using the formula will deepen understanding and navigate through such challenges smoothly.