There are 44 numbered balls in a drum (think lottery). A robot randomly picks 6 balls, I also randomly pick 6 balls. In how many ways can 4 of the balls picked by me be among the 6 balls picked by the robot?

To find the number of ways you can have 4 of your balls among the 6 picked by the robot, we can break it down as follows:

1. **Choose 4 balls from your 6:** This can be done in \( \binom{6}{4} \) ways.
2. **Choose 2 balls from the remaining 38 balls (since 44 total - 6 robot balls = 38 left):** This can be done in \( \binom{38}{2} \) ways.

The total number of ways is given by multiplying these two combinations:

\[
\text{Total Ways} = \binom{6}{4} \times \binom{38}{2}
\]

Calculating each part:

1. \( \binom{6}{4} = 15 \)
2. \( \binom{38}{2} = \frac{38 \times 37}{2} = 703 \)

Now multiply:

\[
\text{Total Ways} = 15 \times 703 = 10455
\]

Thus, there are **10,455 ways** for 4 of your balls to be among the 6 picked by the robot.

Do you want more details or have any questions? Here are some related questions to consider:

1. How would the calculation change if I picked 5 balls instead of 6?
2. What is the probability of having at least 4 balls match?
3. How many ways can you have exactly 2 balls match?
4. If there are additional balls added, how would that affect the outcome?
5. What strategies could increase the likelihood of matching more balls? 

**Tip:** To solve similar problems, always identify how many ways you can select from different groups and use combinations effectively.

Explore the combinatorial problem of matching 4 out of 6 balls in a lottery scenario. This solution demonstrates the calculation of \( \binom{6}{4} \) and \( \binom{38}{2} \), resulting in 10,455 ways for successful matches. Ideal for high school students studying combinatorics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation