Number of ways in which four different toys and five indistinguishable marbles can be distributed between 3 boys, if each boy receives at least one toy and at least one marble

Options

A. 42
B. 100
C. 150
D. 216

Answer

To solve the problem of distributing four different toys and five indistinguishable marbles among three boys such that each boy receives at least one toy and at least one marble, we can break down the solution into two parts: distributing the marbles and distributing the toys.

### Step 1: Distributing the Marbles
1. **Initial Distribution**: Since each boy must receive at least one marble, we start by giving one marble to each of the three boys. This uses up 3 marbles, leaving us with \(5 - 3 = 2\) marbles.

2. **Distributing Remaining Marbles**: Now we need to distribute the remaining 2 indistinguishable marbles among the 3 boys. We can use the "stars and bars" theorem for this distribution.

The formula for distributing \(n\) indistinguishable items into \(r\) distinguishable boxes is given by:
\[ \text{Number of ways} = \binom{n + r - 1}{r - 1} \]
Here, \(n = 2\) (remaining marbles) and \(r = 3\) (boys). Thus, we calculate:
\[ \text{Number of ways} = \binom{2 + 3 - 1}{3 - 1} = \binom{4}{2} = 6 \]

### Step 2: Distributing the Toys
1. **Initial Distribution**: Each boy must receive at least one toy. We can give one toy to each of the three boys. This uses up 3 toys, leaving us with \(4 - 3 = 1\) toy.

2. **Distributing Remaining Toy**: Now we need to distribute the remaining 1 toy among the 3 boys. Since the toys are different, we can choose any of the 3 boys to receive the extra toy. Thus, there are 3 ways to distribute this toy.

### Step 3: Total Combinations
To find the total number of ways to distribute both the marbles and the toys, we multiply the number of ways to distribute the marbles by the number of ways to distribute the toys:
\[ \text{Total ways} = (\text{Ways to distribute marbles}) \times (\text{Ways to distribute toys}) = 6 \times 3 = 18 \]

### Final Answer
The total number of ways to distribute the four different toys and five indistinguishable marbles among three boys, ensuring each boy receives at least one toy and one marble, is **18**.

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