Assume that a particular professional baseball team has 10 ​pitchers, 7 ​infielders, and 13 other players. If 3​ players' names are selected at​ random, determine the probability that all 3 are infielders.

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Part 1

What is the probability that all 3 selected are​ infielders? 

To find the probability that all 3 selected players are infielders, we will follow these steps:

### Step 1: Determine the total number of players.
The team consists of:
- 10 pitchers,
- 7 infielders,
- 13 other players.

Thus, the total number of players on the team is:

\[
10 + 7 + 13 = 30 \text{ players}
\]

### Step 2: Total possible outcomes.
The number of ways to select 3 players from the total of 30 players can be calculated using combinations, denoted as \( \binom{n}{r} \), where \( n \) is the total number of players and \( r \) is the number of players selected. This is given by:

\[
\binom{30}{3} = \frac{30 \times 29 \times 28}{3 \times 2 \times 1} = 4060
\]

### Step 3: Favorable outcomes.
Next, we calculate the number of ways to select 3 infielders from the 7 infielders:

\[
\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

### Step 4: Calculate the probability.
The probability that all 3 selected players are infielders is the ratio of favorable outcomes to total outcomes:

\[
P(\text{all infielders}) = \frac{\binom{7}{3}}{\binom{30}{3}} = \frac{35}{4060} \approx 0.00862
\]

Thus, the probability that all 3 selected players are infielders is approximately **0.00862**, or about **0.86%**.

Would you like further details or have any questions?

---

1. What is the formula for calculating combinations?
2. How does changing the number of infielders affect the probability?
3. What if we were to choose 2 infielders and 1 pitcher?
4. How would the result change if the team had more players?
5. What if 4 players were selected instead of 3?

**Tip:** Always check if the problem involves combinations or permutations by analyzing if the order of selection matters.

Assume that a particular professional baseball team has 10 pitchers, 7 infielders, and 13 other players. If 3 players' names are selected at random, determine the probability that all 3 are infielders.

Determine the probability that all 3 selected players are infielders from a baseball team with 10 pitchers, 7 infielders, and 13 other players. This problem uses the combination formula and basic probability concepts, ideal for students in Grades 9-11.

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