Question

Eleven students are competing in an art contest. In how many different ways can the students finish first, second, and third?

Short answer

There are 990 different ways for the students to finish first, second, and third.

Step-by-step solution

Understand the terminology

Permutations are arrangements of objects in a specific order. In this case, we need to arrange 11 students into 3 different spots (first, second, and third).

Apply the principle of permutation

The principle of permutation states that if there are n different items and we wish to choose r of them (where n > r), then the number of permutations is given by: \(P(n,r) = n * (n - 1) * (n - 2) * ... * (n - r + 1)\). In our case, n = 11 (the number of students) and r = 3 (the number of spots).

Compute the number of permutations

Substituting our n and r values into our formula we get: \(P(11, 3) = 11 * 10 * 9 = 990\). Therefore, there are 990 different ways the students can finish first, second, and third.

Key concepts

Art Contest

An art contest is a competitive event where participants showcase their creativity and artistic skills. In such competitions, individual artists present their works and are judged based on various criteria.
These events are common in schools and allow students to express themselves through different mediums like painting, sculpture, or digital art. They provide a platform for young artists to demonstrate their talents and gain recognition.

Arrangement

Arrangement refers to the way in which things are organized or ordered. In mathematics, particularly in permutations, arrangement is crucial. It determines the possible configurations in which a set of items can be organized.
When it comes to arranging students in an art contest, the arrangement of places (first, second, and third) is essential. Arrangements in permutations consider the specific order in which these positions are filled.

Ordering

Ordering is the process of arranging objects in a sequence. In permutations, ordering is not arbitrary; it follows specific rules that define the sequence. Each order is distinct and counts as a separate possibility.
In our example of an art contest, the ordering of 11 students for the positions of first, second, and third highlights how permutations work. Understanding how to calculate different orderings underpins the solutions to many combination and permutation problems.

Mathematics

Mathematics is the study of numbers, quantities, shapes, and patterns. It provides tools to solve problems involving arrangements and orderings, such as the art contest scenario.
The topic of permutations falls under combinatorics, a branch of mathematics dealing with counting and arrangements. Using formulas like the permutation formula, we solve how many ways participants can be ordered in a competition.

Student Competition

Student competition refers to events where students compete against each other in various fields such as sports, academics, and arts. Competitions foster growth, learning, and personal development.
Art contests are one form of student competition, where individuals can nurture their creativity. These contests not only help in showcasing talents but also prepare students for future challenges by encouraging healthy competition and recognition of skill.