Assume that a particular professional baseball team has 10 ​pitchers, 5 ​infielders, and 10 other players. If 3​ players' names are selected at​ random, determine the probability that 2 are  pitchers and 1 is an infielder.
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Part 1
What is the probability of selecting 2 pitchers and 1​ infielder?

To determine the probability of selecting 2 pitchers and 1 infielder when randomly choosing 3 players from the team, we will use combinations to calculate the total number of possible outcomes and the number of favorable outcomes.

### Step 1: Identify the total number of players and the breakdown
- Total number of players = \(10 \, (\text{pitchers}) + 5 \, (\text{infielders}) + 10 \, (\text{other players}) = 25 \, (\text{players})\)

### Step 2: Calculate the total number of possible ways to select 3 players from 25
The total number of ways to choose 3 players from 25 is given by the combination formula:

\[
\binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300
\]

### Step 3: Calculate the number of favorable outcomes (2 pitchers and 1 infielder)
- The number of ways to choose 2 pitchers from 10 pitchers is:

\[
\binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45
\]

- The number of ways to choose 1 infielder from 5 infielders is:

\[
\binom{5}{1} = 5
\]

Thus, the total number of favorable outcomes (2 pitchers and 1 infielder) is:

\[
\binom{10}{2} \times \binom{5}{1} = 45 \times 5 = 225
\]

### Step 4: Calculate the probability
The probability of selecting 2 pitchers and 1 infielder is:

\[
P(\text{2 pitchers and 1 infielder}) = \frac{225}{2300} \approx 0.0978
\]

Thus, the probability is approximately **0.0978**, or about **9.78%**.

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Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the probability of selecting 3 pitchers at random?
2. How many ways can you select 1 pitcher, 1 infielder, and 1 other player?
3. What is the probability of selecting 2 infielders and 1 pitcher?
4. How would the probability change if there were 15 pitchers instead of 10?
5. What is the probability of selecting 3 players who are not pitchers?

**Tip:** When dealing with probability problems, always double-check the breakdown of categories and ensure you apply combinations correctly to avoid miscalculations.

Assume that a particular professional baseball team has 10 ​pitchers, 5 ​infielders, and 10 other players. If 3​ players' names are selected at​ random, determine the probability that 2 are pitchers and 1 is an infielder.

Learn how to calculate the probability of selecting 2 pitchers and 1 infielder from a baseball team of 25 players using combinations and basic probability rules. This problem demonstrates key combinatorics and probability concepts, ideal for students in Grades 10-12.

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