Problem 75 - Algebra and Trigonometry

Question

Solve the problem using the appropriate counting principle(s). Selecting Prizewinners From a group of 30 contestants, 6 are to be chosen as semifinalists, then 2 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 make these choices in sequence.

Step 1: Choose Semifinalists

We start by selecting 6 semifinalists from a group of 30 contestants. The number of combinations can be calculated using the combination formula, which is given by \( \binom{n}{k} \), where \( n \) is the total number of contestants and \( k \) is the number of semifinalists to choose. So, the number of ways to choose 6 semifinalists from 30 is \( \binom{30}{6} \). Calculating this, we get:\[\binom{30}{6} = \frac{30!}{6!(30-6)!} = 593775\]

Step 2: Choose Finalists

From the 6 semifinalists chosen in step 1, we need to select 2 finalists. Again, we use the combination formula. The number of ways to choose 2 finalists from 6 semifinalists is \( \binom{6}{2} \). Calculating this:\[\binom{6}{2} = \frac{6!}{2!(6-2)!} = 15\]

Step 3: Select the Top Prizewinner

Now, from the 2 finalists, we choose 1 top prizewinner. The number of ways to select 1 prizewinner from 2 finalists is simply 2, since each finalist has an equal chance to be selected as the top prizewinner. Therefore, there are 2 ways to make this selection.

Step 4: Calculate the Total Number of Ways

Finally, to find the total number of ways to make all these choices, we multiply the number of ways to choose semifinalists, finalists, and the prizewinner together. So, the total number of ways is:\[593775 \times 15 \times 2 = 17813250\]

Key Concepts

Combinations

In combinatorics, a combination is a selection of items from a larger set without considering the order of the items. It's a way to count the number of possible ways to select a subset from a collection. For instance, when you choose 6 contestants out of 30, order does not matter, which is why we use combinations. The formula for combinations is given by \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.

For our exercise, selecting 6 semifinalists from 30 contestants is calculated using the formula:

• \( \binom{30}{6} = \frac{30!}{6!(30-6)!} \)
• Which equals 593775 ways to choose the semifinalists.

Combinations are used when the arrangement of the selected items does not matter. This principle significantly reduces the number of possible selections compared to permutations, where order matters.

Factorial

Factorials are a key concept in calculating combinations. A factorial, represented by \( n! \), is the product of all positive integers up to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials are essential in combinatorics as they help determine the number of ways to arrange or select items. In the combination formula, the use of factorials helps control the number of ways different items can be chosen without regard to order. For example:

• In \( \binom{30}{6} = \frac{30!}{6!(30-6)!} \),
• The factorials \( 6! \) and \( (30-6)! \) help account for the redundant permutations of the selected 6 items within 30 contestants.

This ensures that each unique combination is counted just once.

Counting Principles

Counting principles help in determining how many ways certain tasks can be performed by breaking down the task into sequential steps. In our prizewinner selection exercise, multiple steps are needed to carry out the process.

The principles involve:

• Choosing 6 semifinalists from 30, calculated as \( \binom{30}{6} \).
• Selecting 2 finalists from the 6 semifinalists, determined by \( \binom{6}{2} \).
• Picking the top prizewinner from the 2 finalists, simply having 2 ways, as one finalist leads to winning.

The Fundamental Principle of Counting states that if one event can happen in \( m \) ways and a second in \( n \) ways, then both events together can occur in \( m \times n \) ways. This is why we multiplied the number of ways to choose semifinalists, finalists, and the prizewinner to get the final result.

Probability

Probability is a measure of the likelihood of a particular event occurring out of all possible events. Although probability isn't the primary focus of the solution, understanding the principles can be insightful.

In a different context, if you were asked to compute the probability of a specific person being the top prizewinner from the start, you'd need to consider all possible combinations. The total probability would be:

• First, chosen as a semifinalist from 30 \( \to \frac{6}{30} \).
• Then, chosen as a finalist from 6 \( \to \frac{2}{6} \).
• Finally, chosen as the top prizewinner from 2 \( \to \frac{1}{2} \).
• The resulting probability is the product of these fractions: \( \frac{6}{30} \times \frac{2}{6} \times \frac{1}{2} \).

It's crucial to distinguish when to use probability and when counting strategies suffice. This understanding enhances problem-solving skills in various real-world situations.