Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Twelve sprinters are running a heat; those with the best four times will advance to the finals. (a) In how many ways can this group of four be selected? (b) If the four best times will be seeded (ranked) in the finals, in how many ways can this group of four be selected and seeded?

Step-by-Step Solution

Answer

(a) 495 ways; (b) 11,880 ways.

Step 1: Understanding the Problem

We have 12 sprinters and need to select a group of 4 sprinters who will advance to the finals. This part of the problem involves combinations since the order of selection does not matter.

Step 2: Calculating Combinations

The number of combinations can be calculated using the formula for combinations, which is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose. Here, \(n = 12\) and \(r = 4\).

Step 3: Plug Values into Combination Formula

Substitute the values into the combination formula: \[ \binom{12}{4} = \frac{12!}{4! (12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} \].

Step 4: Perform the Calculations for Combinations

Calculate the number of combinations: \[ \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]. So, there are 495 ways to select the group of four sprinters.

Step 5: Understanding Seeding

In part (b), we not only select the sprinters, but also rank them. This involves permutations because the order matters.

Step 6: Calculating Permutations for Seeding

The number of ways to seed the 4 selected sprinters is given by the formula for permutations: \(r!\), where \(r\) is the number of sprinters to be ranked, thus \(4!\).

Step 7: Perform the Calculations for Permutations

Calculate \(4!\): \[4! = 4 \times 3 \times 2 \times 1 = 24\].

Step 8: Combine Combinations and Permutations for Full Solution

The total ways to select and seed the sprinters is given by multiplying the total combinations by the total permutations: \(495 \times 24 = 11880\).

Key Concepts

Permutations

Permutations involve arrangements where the order of items is crucial. Suppose you have a list of items and want to know in how many different sequences you can arrange them. This is where permutations come into play. For example, if you are arranging 4 sprinters for a race, the order in which they run matters, which makes it a permutation problem.

To calculate permutations, we use the formula:

• For all items: \( n! \)
• For selecting \( r \) from \( n \): \( \frac{n!}{(n-r)!} \)

This allows you to determine how many ways you can arrange a subset of items. In our example, once the 4 sprinters are selected, they can be ranked in 24 possible ways since: \( 4! = 24 \).

Permutations are helpful when order or sequence is important, such as rankings or lineups.

Combinations

Combinations help us figure out how many ways we can choose items where the order does not matter. If you are picking teams or groups and don't care about the sequence, combinations are what you need. In problems like our sprinters' example, combinations are used to select 4 out of 12 sprinters to advance to the finals.

The formula for combinations is:

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

Here, \( n \) is the total number of items, and \( r \) is the number of items you are choosing. Plugging the sprinters into the formula gives us:

• \( \binom{12}{4} = \frac{12!}{4!(12-4)!} \)
• This calculates to 495 ways

Combinations are purely about the selection without focus on the arrangement.

Factorial

Factorials are mathematical operations that help calculate permutations and combinations. A factorial is denoted by an exclamation mark (!), and it means you multiply a series of descending whole numbers down to 1.

For example:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Factorials are fundamental to counting arrangements (permutations) and groups (combinations) because they calculate the total possible ways items can be ordered or chosen.

Whenever you see a problem with !, you are dealing with factorials. It simplifies complex calculations in probability, algebra, and more.

Ranking

Ranking refers to ordering individuals or items based on specific criteria. In our exercise, 4 sprinters are not only selected but ranked based on their performance. Ranking is an example of permutations because the sequence in which the sprinters are arranged matters.

When ranking:

• Order is important
• It involves permutations
• 4 sprinters can be ranked in \( 4! = 24 \) ways

Ranking requires a different approach compared to simple selection, as you explore the different possible orderings of a group, often used in competitions or evaluation purposes.

Combining our selection and ranking gives a complete view of the problem, showing just how many possible ways you can choose and arrange contenders.