Question

Thirty contestants, including the local champion, enter a competition. When the first six places are announced: (a) How many different announcements are possible? (b) How many different announcements are possible if the local champion is assured of a place in the first six?

Short answer

(a) There are 30!/(30-6)! ways. (b) There are 29!/(29-5)! ways with the champion included.

Step-by-step solution

Understanding the Context

We have a competition with 30 contestants, and we are interested in the number of ways to announce the first six places. This involves arranging six contestants out of 30.

Using Permutations for Part (a)

For part (a), we need to calculate the number of different ways to arrange six out of the 30 contestants. This is a permutations problem, given by the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \] Where \( n = 30 \) and \( r = 6 \).

Calculation of Permutations for Part (a)

Substituting into the formula, we get: \[ P(30, 6) = \frac{30!}{(30-6)!} = \frac{30!}{24!} \] Which simplifies to: \[ 30 \times 29 \times 28 \times 27 \times 26 \times 25 \] The result of this multiplication gives the number of possible announcements.

Using Conditional Permutations for Part (b)

For part (b), we need the local champion to be in the first six, meaning we arrange 5 out of the remaining 29 contestants in combination with the champion. First, choose 5 spots from the remaining 29 contestants, calculated by: \[ P(29, 5) = \frac{29!}{(29-5)!} = \frac{29!}{24!} \] This simplifies to a multiplication of the first five terms of 29!

Final Calculation for Part (b)

The number of ways the local champion can be included is simply the number of ways to arrange the remaining five contestants among the first six contestants: \[ 29 \times 28 \times 27 \times 26 \times 25 \] The result of this multiplication gives the number of announcements where the local champion is assured a place in the first six.

Key concepts

Permutations

Permutations are a fundamental concept in combinatorics, often encountered when dealing with arrangements or sequences of objects. The core idea behind permutations is to determine how many ways a set can be ordered. For example, when you want to find out how many different ways you can line up your 5 books on a shelf, you use permutations. The formula for permutations is given by:\[ P(n, r) = \frac{n!}{(n-r)!} \]where:

• \( n \) is the total number of objects
• \( r \) is the number of objects to arrange

In our competition problem, we used this formula to find how many ways we could choose and order 6 contestants out of 30.

Conditional Permutations

Conditional permutations come into play when there are specific conditions or constraints on a permutation problem. In our example, part (b) of the exercise introduces such a constraint by assuring the local champion a spot in the top six. Under this condition, we first secure a place for the local champion. This leaves us with the task of arranging the remaining 5 positions from the 29 other contestants. The calculation follows a similar logic as in the basic permutation but under the specific condition that one contestant is already chosen. Therefore, we use:\[ P(29, 5) = \frac{29!}{24!} = 29 \times 28 \times 27 \times 26 \times 25 \]By setting aside the spot for the champion first, we simplify the complex situation into more manageable parts.

Competition Ranking

Competition ranking often involves permutations as contestants need to be arranged in order of their placement or ranking. It’s not just about identifying who participates but determining their exact positions. In many competitions, rankings are crucial for awarding prizes and recognizing achievements. For example, when announcing the top six places, each spot can be distinctly filled by a different contestant, causing a specific arrangement of ranks. This problem often involves calculating permutations - to determine all possible orders in which contestants could rank. Announcing different rankings, especially the top positions, takes into account participants' performance and is typically the climax of the competition, determining who excels.

Probability Theory

Probability theory is a branch of mathematics concerned with analyzing random events. While the problem about ranking contestants mainly uses permutations, probability theory often complements it by assessing the likelihood of certain outcomes. Understanding permutations can enrich your grasp of probability because permutations dictate the number of possible successful outcomes. For instance, in a competition, knowing the number of ways contestants can be ranked helps evaluate the chance of a particular ranking occurring. When applied, probability asks questions like, "What is the likelihood the local champion finishes in the top three?" Though not explicitly required in our exercise, blending permutations with probability theory deepens our understanding of competition dynamics.