Question

Describe a real-life situation where the number of possibilities is given by \({ }_5 P_2\). Then describe a real-life situation that can be modeled by \({ }_5 C_2\).

Short answer

For \({ }_5P_2\), a real-life example would be a basketball coach choosing two players from five to take the first and the second shot. For \({ }_5C_2\), a real-life situation could be a project manager selecting two representatives from a team of five people for a meeting.

Step-by-step solution

Example for Permutation \({ }_5P_2\)

Imagine you're a basketball coach and you need to pick two players from 5 available to take the first and the second shot of the game. The order here matters, because the player who goes first might affect the second player psychologically, hence the situation can be modeled by \({ }_5P_2\). The number of different two-player sequences you can choose is 20(\(5*4\)), according to the formula of permutation.

Example for Combination \({ }_5C_2\)

You are a project manager and from a team of 5 people, you need to select 2 to represent your team in a meeting. The order here doesn't matter, because what's important is who are the two selected, not the order they were selected in. The situation can be modeled by \({ }_5C_2\). The number of different pairs of team members you can pick is 10(\(\frac{5*4}{2}\)), according to the formula of combination.

Key concepts

Real-life Applications

In our day-to-day lives, we often face situations where we need to consider different arrangements or selections. Understanding permutations and combinations can be helpful in these scenarios. For instance, let's consider a basketball coach who needs to choose two players from a group of five to take the first two shots of a game. Here, the order of selection is crucial, making it a permutation problem. Another example is a project manager selecting two team members from a group of five for a meeting. In this case, the order doesn't matter, which makes it a combination problem.

Recognizing whether order is significant can help us decide between using permutations or combinations when calculating possibilities. This differentiation not only aids in problem-solving but also enhances decision-making in professional and personal contexts.

Permutation Calculation

Permutations are all about arranging objects where the order is important. In mathematics, the formula for permutations is expressed as:

• \[ _nP_r = \frac{n!}{(n-r)!} \]

Here, "n" represents the total number of objects, and "r" is the number of objects to choose.

For instance, if we need to determine the number of ways to arrange 2 players out of 5, the exercise provides an example with \(_5P_2\). The calculation follows:

• First, calculate the factorial of 5 (5!), which is 120.
• Next, compute (5-2)!, which is 3! (equals to 6).
• Finally, divide 120 by 6, giving us 20 possibilities where order matters.
  
  This formula helps us systematically determine the number of possible permutations in various real-life scenarios, whether it's selecting players or planning an event schedule where sequence matters.

Combination Calculation

Unlike permutations, combinations are used when the arrangement or order does not matter. This concept is calculated using the formula:

• \[ _nC_r = \frac{n!}{r!(n-r)!} \]

The same letters "n" and "r" denote the total number of available items and those being chosen, respectively.

In our example, if a manager needs to choose 2 team members from a group of 5 for a meeting, we use \(_5C_2\). The steps to calculate this are:

• First, calculate 5!, which is 120.
• Next, calculate 2!, which is 2, and (5-2)!, which is 3!, equalling 6.
• Finally, the formula simplifies to \( \frac{120}{2 \times 6} \) = 10 possible combinations.
  
  By using combinations, we focus on the elements selected rather than the sequence. This distinction is vital in situations such as forming committees or choosing groups, where only the selection itself matters.

Probability Theory

Probability theory deals with the likelihood of occurrences and often utilizes concepts of permutations and combinations. When calculating probabilities, understanding whether to use permutations or combinations is key to getting accurate results.

For example, suppose you want to find the chances of picking any two players out of five randomly to perform a specific task without concern for order. Here, combinations inform the "favorable outcomes," which are calculated first, followed by the ratio against total possible outcomes.

Probability is broadly defined by the formula:

• Probability = \( \frac{ \text{Number of favorable outcomes} }{ \text{Total number of possible outcomes} } \)

Thus, effectively calculating permutations and combinations not only assists in probability calculations but deepens our comprehension of random events, enabling more informed predictions and assessments of chance in diverse fields such as gaming, risk analysis, and beyond.