30 students are trying out for the new co-ed BCSS soccer team. There are 3 goalies, 9 strikers,  12 midfielders, and 6 defenders. A team of 20 players will be chosen. What is the probability at least two strikers will be chosen?

To calculate the probability that **at least two strikers** will be chosen from the group of 30 students trying out for the soccer team, we need to break the problem into a few steps.

### Step 1: Total number of ways to select the team
The number of ways to choose 20 players from 30 students is calculated using combinations:

\[
\binom{30}{20}
\]

This represents the total number of possible ways to select 20 players from 30.

### Step 2: Unfavorable scenarios (fewer than two strikers)
We need to calculate the number of teams that contain **fewer than two strikers**, i.e., either 0 or 1 striker.

#### Case 1: No strikers
If no strikers are chosen, we must choose all 20 players from the 21 remaining players (which includes the 3 goalies, 12 midfielders, and 6 defenders):

\[
\binom{21}{20} = 21
\]

#### Case 2: One striker
If exactly one striker is chosen, we must choose the other 19 players from the 21 non-strikers (goalies, midfielders, and defenders):

\[
\binom{9}{1} \times \binom{21}{19}
\]

Breaking this down:
- \(\binom{9}{1} = 9\) (ways to choose 1 striker)
- \(\binom{21}{19} = \binom{21}{2} = \frac{21 \times 20}{2} = 210\)

So the total for this case is:

\[
9 \times 210 = 1890
\]

### Step 3: Total unfavorable outcomes
Adding the two cases together gives the total number of unfavorable outcomes (where fewer than two strikers are chosen):

\[
21 + 1890 = 1911
\]

### Step 4: Total favorable outcomes
The total number of ways to choose 20 players is \(\binom{30}{20}\). To compute this, we use the property \(\binom{30}{20} = \binom{30}{10}\) (since \(\binom{n}{r} = \binom{n}{n-r}\)):

\[
\binom{30}{10} = \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25 \times 24 \times 23 \times 22 \times 21}{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}
\]

This calculation equals **300,450**.

### Step 5: Probability calculation
The probability that at least two strikers are chosen is the complement of the probability that fewer than two strikers are chosen. So the probability is:

\[
P(\text{at least two strikers}) = 1 - \frac{1911}{300450}
\]

Now let's calculate this:

\[
P(\text{at least two strikers}) \approx 1 - 0.00636 = 0.99364
\]

So, the probability is approximately **0.99364** or **99.36%**.

### Conclusion:
The probability that at least two strikers will be chosen is approximately **99.36%**.

---

Would you like more details or have any questions?

Here are some related questions to consider:
1. How would the probability change if the team size were smaller?
2. What if the number of strikers trying out increased?
3. How would the scenario change if exactly three strikers must be chosen?
4. What if the team required a specific number of goalies?
5. How do the combinations change if additional restrictions are placed on midfielders?

**Tip**: When calculating large combinations, use the property \(\binom{n}{r} = \binom{n}{n-r}\) to simplify calculations.

30 students are trying out for the new co-ed BCSS soccer team. There are 3 goalies, 9 strikers, 12 midfielders, and 6 defenders. A team of 20 players will be chosen. What is the probability at least two strikers will be chosen?

Calculate the probability that at least two strikers will be chosen from 30 soccer players, including goalies, midfielders, and defenders. This detailed solution covers combinatorics and probability theory, providing a step-by-step breakdown of unfavorable cases and complementary probability. Suitable for students in Grades 10-12.

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