Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Eight sprinters have qualified for the finals in the 100 -meter dash at the NCAA national track meet. How many different orders of the top three finishes are possible? (Assume there are no ties.)

Step-by-Step Solution

Answer

The number of different possible orders for the top three finishes among the eight sprinters is 336.

Step 1: Identify the values of n and r

In this exercise we have eight sprinters and we want to know the number of orders for the top three places. Thus, the number of total items \(n\) is 8 and the number of items we are selecting \(r\) is 3.

Step 2: Apply the permutation formula

We will use the permutation formula \(P(n,r) = n! / (n-r)!\) to calculate the number of possible orders. Substituting the known values: \(P(8,3) = 8! / (8-3)!\).

Step 3: Calculate factorial and solve

First calculate the factorial of the numbers: \(8! = 8*7*6*5*4*3*2*1\) and \(5! = 5*4*3*2*1\). Then substitute into the permutation equation and perform the division. The final result is the number of possible orders for the top three finishes.

Key Concepts

Factorial

The concept of factorial is pivotal in understanding permutations and combinations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number. For example, to find 8 factorial, written as \(8!\), we multiply all whole numbers from 1 to 8 together:

• 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.

This results in 40,320. Factorials are used extensively in permutations because they allow us to calculate all possible arrangements of a set of items.
For instance, if you want to know how many ways you can arrange 8 distinct sprinters, you'd calculate \(8!\).
For our exercise, the calculation needed was \(P(8,3)\), which equates to dividing \(8!\) by \(5!\). This is due to the fact that we only care about the top 3 places, so we exclude the arrangements of the 5 sprinters who are not in those top spots.

Combinatorics

Combinatorics is a field of mathematics focused on counting and arranging items. It's like the art of figuring out how things can be ordered or chosen in diverse ways. Two primary concepts within combinatorics are permutations and combinations.

• Permutations: These are arrangements where the order matters. The formula for permutations is \(P(n, r) = \frac{n!}{(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items you want to arrange.
• Combinations: Here, the order does not matter. The formula for combinations is different and not used in our track meet problem, but it's worth noting for other contexts.

Our exercise required permutations because we were interested in the different "orders" of sprinters. The order of finishing matters to determine who gets the gold, silver, or bronze.
Understanding permutations helps solve such arrangement problems efficiently.

NCAA track meet

An NCAA track meet is a championship event where athletes from different colleges compete in various track and field events. Imagine the electrifying atmosphere as sprinters line up for the finals in a 100-meter dash. In these finals, it's not just about participating but finishing in one of the top spots.

• The top three sprinters gain medals: gold for first, silver for second, and bronze for third. Thus, determining every possible order of these top three finishes is crucial.
• With 8 sprinters racing, multiple combinations can determine who stands on the podium, exactly \(P(8,3)\).

By applying what we learned about permutations, we know there are several sequences for the top three finishes, considering no ties occur.
Each sequence could be a different exciting outcome for the competitors involved.