Problem 71 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Selecting Prizewinners From a group of 30 contestants, six are to be chosen as semifinalists, then two of those are chosen as finalists, and then the top prize is awarded to one of the finalists. In how many ways can these choices be made in sequence?

Step-by-Step Solution

Answer

There are 17,813,250 ways to choose the winners in sequence.

Step 1: Choose Semifinalists

First, we select six semifinalists from the group of 30 contestants. The number of ways to do this is given by the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \). Here, \( n = 30 \) and \( k = 6 \). Calculate \( C(30, 6) = \frac{30!}{6!(30-6)!} \).

Step 2: Calculate Combinations for Semifinalists

Calculate \( C(30, 6) = \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 593775 \).

Step 3: Choose Finalists from Semifinalists

Once the semifinalists are chosen, select 2 finalists from the 6 semifinalists. The number of ways to do this is \( C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \).

Step 4: Calculate Ways to Select the Top Prize Winner

From the 2 finalists, choose 1 to receive the top prize. The number of ways to choose the top prize winner is \( C(2, 1) = 2 \).

Step 5: Multiply the Number of Ways

Multiply the number of ways to select the semifinalists, finalists, and top prize winner: \( 593775 \times 15 \times 2 = 17813250 \). This is the total number of ways to choose the winners in sequence.

Key Concepts

Combination Formula

When it comes to selecting a group from a larger set, the combination formula is your best friend. It helps you figure out the number of ways to choose a specific number of items from a larger pool, without worrying about the arrangement. For example, when choosing six semifinalists from 30 contestants, the order in which you pick them doesn't matter. That's where combinations come into play.

The combination formula is represented as:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

Here, "\( n \)" is the total number of items to choose from, and "\( k \)" is the number of items to choose. The "!" symbol stands for factorial, which means you multiply a series of descending natural numbers. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Using this formula, you can calculate how many different groups can be formed. It's a fundamental concept in combinatorics, focusing on the subset of possibilities without concern for the order.

Counting Principle

The counting principle is a straightforward yet powerful tool for determining the number of possible outcomes when you're dealing with multiple choices or stages. The principle states that if you have to make a decision in sequence, the total number of ways to achieve a result is the product of the number of ways each decision can be made.

For example, in our problem:

• First, there are 593,775 ways to choose the six semifinalists from 30 contestants.
• Then, from these six, there are 15 ways to choose two finalists.
• Finally, there are 2 ways to select one top prize winner from the finalists.

According to the counting principle, you multiply these numbers together to find the total number of sequences possible, which is 17,813,250. This principle helps streamline complex probability questions into simpler, multiplicative solutions.

Permutations

Permutations focus on the arrangement of a set of items where the order does matter. It's different from combinations because in permutations, swapping the position of items results in a different outcome. While the given exercise doesn't directly involve calculating permutations, understanding this concept is crucial when the arrangement sequence is important, such as assigning ranks or positions.

The formula for permutations is:

• \( P(n, k) = \frac{n!}{(n-k)!} \)

For example, choosing a lead and an assistant from a team where roles differ uses permutations. Remember, permutations are key whenever each arrangement counts as a unique result. This would be the next step if we needed to rank the participants instead of simply selecting them.

Probability

Probability is the measure of the chance of a particular outcome occurring in a random experiment. It operates on the premise of predicted outcomes divided by the total number of possible outcomes. While probability isn't overtly calculated in the exercise, every step of determining ways to choose winners contributes to understanding possible outcomes.

To connect it to our example, if all possible outcomes have been correctly identified, probability can help ascertain the chance of ending up with a specific set of winners. However, probability usually requires a defined set of outcomes, as in picking certain winning contestants randomly. Beyond this, probability helps in making informed predictions and decisions in uncertain conditions.

The basic formula for probability is:

• \( P(E) = \frac{Number\ of\ Favorable\ Outcomes}{Total\ Number\ of\ Possible\ Outcomes} \)

Understanding probability aids in gauging the likelihood of certain selections in real-life scenarios.