Number of ways in which n balls be randomly distributed in n cells is

Options

A. `n!`
B. `n(n+1)`
C. `n^(n)`
D. `2^(n)`

Answer

To solve the problem of finding the number of ways in which \( n \) balls can be randomly distributed in \( n \) cells, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Balls and Cells**:
We have \( n \) balls and \( n \) cells.

2. **Distribution of the First Ball**:
The first ball can be placed in any of the \( n \) cells. Therefore, there are \( n \) choices for the first ball.

3. **Distribution of the Second Ball**:
The second ball can also be placed in any of the \( n \) cells. Since it is not specified that a cell can only contain one ball, the second ball also has \( n \) choices.

4. **Continue for All Balls**:
This reasoning continues for all \( n \) balls. Each ball has \( n \) choices of cells.

5. **Calculate the Total Number of Ways**:
Since each of the \( n \) balls has \( n \) choices, the total number of ways to distribute the balls is given by multiplying the number of choices for each ball:
\[ \text{Total Ways} = n \times n \times n \times \ldots \text{(n times)} = n^n \]

6. **Conclusion**:
Thus, the total number of ways in which \( n \) balls can be randomly distributed in \( n \) cells is \( n^n \).

### Final Answer:
The number of ways in which \( n \) balls can be randomly distributed in \( n \) cells is \( n^n \).

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