Math: Permutation, Combination, and Probability Quiz

Math: Understanding Permutation, Combination and Probability

Introduction

Math involves various concepts that are crucial to understanding the world around us. Three fundamental subtopics within math are permutation, combination, and probability. These topics help us calculate probabilities using different methods based on the context of the problem. In this article, we will delve into these three subtopics and explore how they are related.

Permutation

Permutation is the arrangement of items in a specific order. For example, if there are 10 books and you want to arrange them on a shelf, each arrangement is considered a unique permutation. Calculating a permutation involves finding the factorial of a number. If there are n items to choose from, the total number of permutations would be n!, where ! denotes the factorial function.

Example

Suppose you have 10 students called Alice, Bob, Carlos, Dave, Eve, Frank, Grace, Harry, Ian, and Josh. If you need to give out five awards, calculating permutations with repetition means you can award more than one student for the same category. You could use the formula P(n, r) = n!(n-r+1), where n represents the total number of students and r represents the number of awards. In this case, P(10, 5) = 10!/(10-5+1) = 30,240 possible arrangements.

Combinations

Combinations involve selecting items without considering their order. Returning to our book example, combinations would mean choosing any four books out of the 10 available without caring about their position. The formula for calculating combinations is C(n, k) = n!/((n-k)!*k!) * (k/n), where n represents the total number of items and k represents the number of items chosen.

Example

Using combinations, we could find out how many ways we can form a committee of three people from 10 friends. The calculation would be C(10, 3) = 10!/((10-3)!*3!)*(3/10) = 120 possible committees.

Probability

Probability is the likelihood of an event occurring. By combining the principles of permutations and combinations, probability can be calculated by dividing the favorable outcomes by the total possible outcomes. For example, if there are 10 items to choose from and we want to select two items, the probability of drawing exactly two red balls from a deck of cards containing 52 cards can be calculated as follows:

• Total possible outcomes = 10!/(10-2+1) = 45 possible pairings of two books
• The number of favorable outcomes (drawing exactly two red balls) is given by the formula for combinations: C(n, k), where n represents the total number of items and k represents the desired outcome. In this case, n=13 (two red balls and one black ball), and k=2 (the desired outcome): C(13, 2) = 65 possible outcomes of drawing exactly two red balls.

Example

Suppose we have a deck of playing cards with four suits, each suit having three face cards: spades (clubs), hearts, diamonds, and clubs. We want to find the probability of being dealt a hand consisting of one face card from each suit (for example, 3, 4, 5 of spades, hearts, diamonds, and clubs). The calculation would be:

• Total possible outcomes = 48 (total number of cards in a standard 52-card deck)
• Favorable outcomes = C(12, 1)*C(12, 1)*C(12, 1)*C(12, 1) = 479001600 possible hands with one face card from each suit

So, the probability of getting such a hand is 479001600/48 = 997916800/48 * 1/1 = 997916800/48 * (1/1) = 997916800/48 * 1 = 21600/48 = 4458.3333333333336.

In conclusion, permutation, combination, and probability are essential subtopics in math that help us understand and calculate probabilities. These concepts are interconnected, and by understanding how they work together, we can tackle a wide range of problems and situations in various fields.