Question

The organizer of a cast party for a drama club asks each of the 6 cast members to bring 1 food item from a list of 10 items. Assuming each member randomly chooses a food item to bring, what is the probability that at least 2 of the 6 cast members bring the same item?

Short answer

The probability that at least 2 of the 6 cast members bring the same food item is given by \( P = N_{at least 1 duplicate} / N_{total} \), where \( N_{at least 1 duplicate} \) and \( N_{total} \) are calculated in Step 4 and Step 2 respectively. Without having specific calculations, the exact answer can't be provided here.

Step-by-step solution

Understand the problem

The question asks for the probability that at least 2 of the 6 cast members bring the same item. The total number of outcomes is the number of ways that 10 different food items can be picked by 6 people. Each person can pick any of the 10 food items, regardless of what the others pick, so the total number of outcomes, \( N_{total} \), is \( 10^6 \).

Calculate the total number of outcomes

Each person can choose any of the 10 items, and there are 6 people, so there are \( 10^6 \) possible ways to assign an item to each person. Using the formula of permutation without replacement, we can calculate each unique set of food items that can be brought by the six drama club members and we will find out that the total number of outcomes is 10 raised to the power of 6.

Calculate the number of outcomes with no duplicates

Now, it's time to find out the number of outcomes where no two members bring the same food item. This is a permutation without replacement for the 6 people selecting the food items. So, \( N_{no duplicates} \) = \( P(10, 6) \) = \( 10 * 9 * 8 * 7 * 6 * 5 \).

Calculate the number of outcomes with at least 1 duplicate

With the two previous steps, it's possible to calculate the number of outcomes which must involve at least 1 duplicated food item. This is \( N_{at least 1 duplicate} \) = \( N_{total} - N_{no duplicates} \).

Calculate the probability

Now, we can compute the probability that at least two members choose the same food item, which is the number of favourable outcomes over the total number of outcomes, \( P \) = \( N_{at least 1 duplicate} / N_{total} \).

Key concepts

Permutations

Permutations reflect the different ways to arrange a set of items or elements. When dealing with permutations, order is crucial. In the context of our problem, we want to know how many different ways 10 food items can be brought by 6 different cast members, ensuring no repeats in food choice.

If we consider permutations without replacement, this means once an item is chosen, it cannot be picked again. The formula for this permutation is denoted as \( P(n, r) \), which calculates the number of ways to arrange \( r \) selections from \( n \) items, like \( P(10, 6) \) in the exercise.

This leads to the calculation of \( 10 \times 9 \times 8 \times 7 \times 6 \times 5 \), because:

• First choice: 10 options
• Second choice: 9 options (as one is already taken)
• Continues until 6 choices are made

Using permutations helps in determining outcomes, especially in problems where the uniqueness of each choice is important.

Combinatorics

Combinatorics is the field of mathematics that focuses on counting, arranging, and analyzing configurations or structures formed by a finite set of items. It's the underlying math that supports permutations and combinations.

In scenarios like our problem, combinatorics helps in understanding not just single arrangements but also how multiple criteria or conditions affect possible outcomes. Here’s how it applies to our exercise:

• It calculates how different combinations of choices can lead to unique or repeated selections.
• This includes determining which configurations cause overlaps or conflicts, such as duplicated food item selections.

Combinatorics is the backbone allowing us to systematically tackle problems that seem like they have too many possibilities to approach directly.

Probability Theory

Probability theory deals with the likelihood of different outcomes. In our exercise, the goal was to find the probability that at least two of the 6 cast members end up with the same food item.

Here's how probability theory steps in:

• First, calculate the total number of possible outcomes (e.g., each person can bring any of 10 items), leading to \( 10^6 \).
• Next, determine the outcomes where no items are repeated using permutations (as discussed).
• Subtract these non-repeating outcomes from the total, giving outcomes with at least one duplicate.
• Finally, finding the probability involves dividing the above discreet outcomes by the total possible outcomes.

This systematic approach breaks down complex random occurrences into simpler probabilities and enables us to quantify the likelihood of specific events.