125 students have volunteered to be on the prom committee. 2 of those students will be randomly chosen to win free tickets to the prom. How many different ways can the winners be chosen? 1. Does order matter? 2. This is a 3. How many different ways can winners be chosen? 4. What is the probability that the first two students who signed up will be the winners? P(1st two students who signed up win) =

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UpStudy AI · Accepted Answer

Certainly! Let's tackle each of your questions step by step related to selecting winners from the prom committee.

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**1. Does Order Matter?**

**Answer:** **No, order does not matter.**

**Explanation:**  
When selecting winners for prizes (like free tickets to the prom), the arrangement in which the winners are chosen doesn't affect the outcome. Whether Student A is chosen first and Student B second is the same as selecting Student B first and Student A second. Therefore, this is a **combination** problem, not a permutation.

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**2. This is a**

**Answer:** **Combination problem.**

**Explanation:**  
As established in the first question, since the order of selection doesn't matter, you're dealing with combinations. Specifically, you're choosing 2 students out of 125 without regard to the order in which they're selected.

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**3. How Many Different Ways Can Winners Be Chosen?**

**Answer:**  
The number of different ways to choose the winners is given by the combination formula:

$$
C(n, k) = \frac{n!}{k!(n - k)!}
$$

Where:
- $$ n = 125 $$ (total students)
- $$ k = 2 $$ (winners to choose)

Calculating:

$$
C(125, 2) = \frac{125!}{2!(125 - 2)!} = \frac{125 \times 124}{2 \times 1} = 7,750
$$

**So, there are 7,750 different ways to choose 2 winners from 125 students.**

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**4. What is the Probability That the First Two Students Who Signed Up Will Be the Winners?**

**Answer:**  
To find this probability, consider the total number of ways to choose 2 winners out of 125, which we've already determined is 7,750.

Since there's only **1 favorable outcome** where the first two specific students are chosen, the probability $$ P $$ is:

$$
P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{1}{7,750}
$$

**So, the probability that the first two students who signed up will win is $$ \frac{1}{7,750} $$.**

Alternatively, you can compute it step by step:

1. **Probability the first student is selected:**  
   $$
   \frac{2}{125}
   $$
   (since there are 2 winning spots)

2. **Probability the second student is selected given the first was already selected:**  
   $$
   \frac{1}{124}
   $$

3. **Multiply the two probabilities:**  
   $$
   \frac{2}{125} \times \frac{1}{124} = \frac{2}{15,500} = \frac{1}{7,750}
   $$

**Thus, $$ P(\text{1st two students who signed up win}) = \frac{1}{7,750} $$.**

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If you have any more questions or need further clarification on any of these points, feel free to ask!
**1. Does order matter?** No, order does not matter. **2. This is a** Combination problem. **3. How many different ways can winners be chosen?** There are 7,750 different ways to choose the winners. **4. What is the probability that the first two students who signed up will be the winners?** The probability is $$ \frac{1}{7,750} $$.

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