We have 3 balls and 4 cups. We randomly place the balls into the cups. We want to find the probabilities that the maximum number of balls in a single cup is 1, 2, and 3, respectively.

1. Problem Description

We have 3 balls and 4 cups. We randomly place the balls into the cups. We want to find the probabilities that the maximum number of balls in a single cup is 1, 2, and 3, respectively.

2. Solution Steps

First, we find the total number of ways to place the 3 balls into 4 cups. Each ball has 4 choices of which cup to go into. Since the balls are placed independently, the total number of ways is $4 \times 4 \times 4 = 4^3 = 64$.

Case 1: The maximum number of balls in a single cup is

1. This means each cup has at most 1 ball. We are choosing 3 cups out of 4 to put the balls in, and we assume the balls are distinguishable (say, numbered 1, 2, and 3).

So, we choose 3 cups out of 4, and then assign the balls to the cups. The number of ways to choose 3 cups out of 4 is $\binom{4}{3} = 4$. Then, there are $3! = 6$ ways to put the 3 balls into the 3 chosen cups. So, there are $4 \times 6 = 24$ ways.

The probability is $\frac{24}{64} = \frac{3}{8}$.

Case 2: The maximum number of balls in a single cup is

2. This means one cup has 2 balls and another cup has 1 ball.

First, choose which 2 balls go together, which is $\binom{3}{2} = 3$ ways. Then choose which cup these 2 balls go into. There are 4 choices for the cup containing two balls. Then choose which of the remaining 3 cups the last ball goes into. So there are 3 choices. The total number of ways is $3 \times 4 \times 3 = 36$.

The probability is $\frac{36}{64} = \frac{9}{16}$.

Case 3: The maximum number of balls in a single cup is

3. This means one cup has 3 balls.

First, choose which cup these 3 balls go into. There are 4 choices for the cup. The total number of ways is

4. The probability is $\frac{4}{64} = \frac{1}{16}$.

The sum of the probabilities should be

1. $\frac{3}{8} + \frac{9}{16} + \frac{1}{16} = \frac{6}{16} + \frac{9}{16} + \frac{1}{16} = \frac{16}{16} = 1$.

3. Final Answer

The probabilities are $\frac{3}{8}$, $\frac{9}{16}$, and $\frac{1}{16}$.