Solve by the method of your choice.

Twenty dash threeTwenty-three 

people purchase raffle tickets. Three winning tickets are selected at random. If first prize is 

​$20002000​, 

second prize is 

​$15001500​, 

and third prize is 

​$200200​, 

in how many different ways can the prizes be​ awarded?

Question content area bottom

Part 1

There are 

enter your response here 

different ways in which the prizes can be awarded.

​(Simplify your​ answer.)

To solve this problem, we need to determine the number of ways to assign three different prizes to three winners selected from a group of 23 people. 

### Understanding the Problem

We are selecting three winners out of 23 people and assigning each winner a specific prize. The prizes are:
- First prize: $2000
- Second prize: $1500
- Third prize: $200

The order in which the winners are chosen matters because each prize is different. This is a permutation problem where we are concerned with the arrangement of winners.

### Step-by-Step Calculation

1. **Select the First Prize Winner:**
   - There are 23 people eligible for the first prize.
   - Thus, there are 23 choices for the first prize winner.

2. **Select the Second Prize Winner:**
   - After selecting the first prize winner, there are 22 people left.
   - Thus, there are 22 choices for the second prize winner.

3. **Select the Third Prize Winner:**
   - After selecting the first and second prize winners, there are 21 people left.
   - Thus, there are 21 choices for the third prize winner.

4. **Calculate the Total Number of Ways:**
   - The total number of ways to award the prizes is the product of the choices for each winner.
   
\[
\text{Total ways} = 23 \times 22 \times 21
\]

Let's calculate this:

\[
23 \times 22 = 506
\]
\[
506 \times 21 = 10626
\]

### Conclusion

There are **10,626** different ways in which the prizes can be awarded.

---

Would you like more details on any of these steps, or do you have any additional questions?

### Related Questions:

1. How would the number of ways change if there were four different prizes instead of three?
2. What if the prizes were identical and we only needed to choose three winners?
3. How does the number of ways change if there were 30 people instead of 23?
4. If the prizes were $3000, $1500, and $500 respectively, how does this affect the number of ways they can be awarded?
5. What if each prize could be awarded to the same person multiple times?

**Tip:** In problems involving permutations, order matters. This is different from combinations where the order does not matter. Make sure to identify which type of counting is required for the problem you're solving.

Discover how to calculate the number of ways to award prizes when selecting winners from a group using permutations. Solve the problem of distributing prizes of $2000, $1500, and $200 among 23 participants with step-by-step calculations.

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