Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

The graphic design department is holding a contest in which it will award scholarships of different values to those who finish in first place, second place, and third place. The department receives 104 entries. How many different orders of the top three places are possible?

Step-by-Step Solution

Answer

There are 1,091,016 different arrangements of the top three places possible.

Step 1: Understand the Permutations formula

The Permutations formula is given by \(P(n, r) = n! / (n-r)!\) where \(n\) is the total number of items, and \(r\) is the number of places or spots to fill. This formula is used when the order of the items or objects matters. In this case, we are trying to find out the number of ways to arrange 104 contestants into the top three spots, so the order does matter.

Step 2: Substitute the values into the Permutations formula

We substitute \(n = 104\) and \(r = 3\) into the formula. This gives us \(P(104, 3) = 104! / (104-3)!\)

Step 3: Calculate the factorial and simplify

Calculate the factorial values and simplify the expression. We get \(P(104, 3) = 104*103*102\) After calculation, we find that \(P(104, 3) = 1,091,016\)

Step 4: Conclusion

There are 1,091,016 different orders of the top three places possible from the total of 104 entries.

Key Concepts

Factorial

The concept of a factorial is foundational in the study of permutations and combinations. A factorial, represented by an exclamation mark (!), refers to the product of all positive integers up to a given number. For example, the factorial of 4 (written as \(4!\)) is the product of 4, 3, 2, and 1, which equals 24.

Here is a breakdown of how factorials grow with each additional number:

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

Factorials are a key component in arranging objects because they express the number of ways to arrange a certain number of distinct items into a sequence.

Permutation Formula

When we need to find the number of possible ways to arrange a set number of objects in a particular order, the permutation formula is used. The general permutation formula is \(P(n, r) = \frac{n!}{(n-r)!}\), where \(n\) is the total number of items to choose from, and \(r\) is the number of items to arrange.

To understand the permutation formula better, suppose we have three books and want to know how many ways we can arrange two of them on a shelf. Using the formula, with \(n = 3\) and \(r = 2\), we have \(P(3, 2) = \frac{3!}{(3-2)!} = \frac{3!}{1!} = \frac{6}{1} = 6\). Therefore, there are six different ways to arrange two out of three books.

Arrangement of Objects

The arrangement of objects is essentially the act of putting items in a particular order. In mathematics, this is a permutation problem where order matters. When one speaks of arranging objects, it is implied that each arrangement, or permutation, is unique from the others, based on the order of the objects.

For instance, if we have three medals: gold, silver, and bronze, arranging them in different orders such as bronze-silver-gold, gold-bronze-silver, or silver-gold-bronze would each count as distinct arrangements. In the original exercise involving 104 entries competing for three top spots, we used the permutation formula to calculate the number of such unique arrangements possible, considering that each placement in the top three is unique and significant in its own right. This is a crucial aspect of combinatorial problems seen in various real-world applications like tournaments, elections, and seating arrangements.