Question

Find the probabilities. \(\frac{C_{2}^{2} C_{1}^{2}}{C_{3}^{4}}\)

Short answer

Answer: The probability is 1.

Step-by-step solution

In order to solve this exercise, we need to understand what combinations are and how to calculate them. Combinations are ways of choosing a subset from a set, without considering the order of the elements. The notation for combinations is \(C_{k}^{n}\), which means choosing k elements from a set of n elements. The formula for calculating combinations is given by:\(C_{k}^{n} = \frac{n!}{k!(n-k)!}\), where n! (n factorial) is the product of all positive integers less than or equal to n and ! denotes factorial. #Step 2: Calculate the combinations in the numerator#

In the given formula, the numerator is \(C_{2}^{2} C_{1}^{2}\). Let's calculate these two combinations: \(C_{2}^{2} = \frac{2!}{2!(2-2)!} = \frac{2!}{2!0!} = \frac{2}{1} = 2\) \(C_{1}^{2} = \frac{2!}{1!(2-1)!} = \frac{2!}{1!1!} = \frac{2}{1} = 2\) Now, we multiply these two combinations together: \(C_{2}^{2} C_{1}^{2} = 2 \times 2 = 4\) #Step 3: Calculate the combination in the denominator#

In the given formula, the denominator is \(C_{3}^{4}\). Let's calculate this combination: \(C_{3}^{4} = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = \frac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4\) #Step 4: Calculate the probability#

Now that we have calculated the numerator and denominator of the given formula, we can find the probability: Probability = \(\frac{C_{2}^{2} C_{1}^{2}}{C_{3}^{4}} = \frac{4}{4} = 1\) Therefore, the probability is 1.

Key concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, arranging, and finding patterns in different configurations of objects regardless of their order. It is the backbone of probability and various other fields of mathematics and computer science, as it provides the principles needed to compute quantities and manage permutations and combinations.

When we look at combination problems, we are often interested in finding out how many ways we can choose a subset of items from a larger set. A common example would be choosing a committee from a larger group—it doesn't matter the order in which committee members are selected, only who is on the committee. This is where the concept of combinations as distinct from permutations emerges, as permutations do take the order into consideration.

Factorial Notation

Factorial notation is a mathematical concept used to describe the product of a series of descending natural numbers and is denoted with an exclamation mark (e.g., 5!, pronounced 'five factorial'). For any non-negative integer n, its factorial is the product of all positive integers less than or equal to n, defined by the equation:
\[\begin{equation} n! = n \times (n - 1) \times (n - 2) \times \text{...} \times 1 \text{with the special case} 0! = 1. \text{This means} 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.\text{In combinatorics, factorial notation is crucial in calculating permutations and combinations, as seen in the combination formula}\( C_{k}^{n} = \frac{n!}{k!(n-k)!} \).\text{It's used to evaluate the total number of distinct groupings (combinations) without regard to order.}

Probability Calculation

Probability calculation involves determining the chance or likelihood that a certain event will happen. It is a fundamental concept in statistics, gaming, statistical mechanics, and much more. Probabilities are given a value between 0 (the event will definitely not happen) and 1 (the event will definitely happen).

Mathematically, the probability of an event is expressed as the ratio of the number of favorable outcomes to the total number of possible outcomes. For instance, the chance of rolling a '3' on a fair six-sided die is 1 out of 6, or approximately 0.167. When dealing with combinations, the probability of selecting a specific combination is calculated by dividing the number of ways to arrange the desired subset (favorable outcomes) by the total number of ways to arrange any subset (possible outcomes).