A child has five pockets and seven marbles. In how many ways can the child put the marbles in his pockets?

Explanation

To solve the problem of how many ways a child can put 7 marbles into 5 pockets, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- We have 5 pockets (let's denote them as P1, P2, P3, P4, P5).
- We have 7 marbles (let's denote them as M1, M2, M3, M4, M5, M6, M7).

2. **Determine the Options for Each Marble**:
- For each marble, there are 5 choices (it can go into any of the 5 pockets).

3. **Calculate the Total Choices**:
- Since the choice for each marble is independent of the others, we can multiply the number of choices for each marble.
- For the first marble (M1), there are 5 options.
- For the second marble (M2), there are also 5 options.
- This pattern continues for all 7 marbles.

4. **Formulate the Expression**:
- The total number of ways to distribute the marbles into the pockets can be expressed as:
\[ 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^7 \]

5. **Calculate \(5^7\)**:
- Now, we need to calculate \(5^7\).
- \(5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78,125\).

6. **Final Answer**:
- Therefore, the total number of ways the child can put the marbles in his pockets is **78,125**.