The number of ways in which we can put 5 different balls in 5 different boxes such that atmost three boxes are empty, is equal to

Options

A. `5^(5)+5`
B. `5^(5)-10`
C. `5^(5)-5`
D. `5^(5)-4^(5)`

Answer

To solve the problem of finding the number of ways to put 5 different balls into 5 different boxes such that at most 3 boxes are empty, we can follow these steps:

### Step 1: Calculate Total Arrangements
First, we calculate the total number of arrangements without any restrictions. Each of the 5 balls can go into any of the 5 boxes. Therefore, the total number of ways to arrange the balls in the boxes is given by:

\[ 5^5 \]

### Step 2: Calculate Arrangements with 4 Empty Boxes
Next, we need to consider the cases where 4 boxes are empty. If 4 boxes are empty, then only 1 box is occupied. We can choose which 1 box will be occupied in:

\[ \binom{5}{1} = 5 \text{ ways} \]

Once we have chosen the box, all 5 balls must go into that box. Thus, there is only 1 way to place the balls in the chosen box. Therefore, the total number of arrangements with 4 empty boxes is:

\[ 5 \times 1 = 5 \]

### Step 3: Calculate Arrangements with 5 Empty Boxes
It is impossible to have all 5 boxes empty if we are required to place at least one ball. Therefore, the number of arrangements with 5 empty boxes is:

\[ 0 \]

### Step 4: Combine the Results
Now, we can combine the results to find the total number of arrangements where at most 3 boxes are empty. We subtract the cases where 4 or more boxes are empty from the total arrangements:

\[ \text{Total arrangements with at most 3 empty boxes} = 5^5 - \text{(arrangements with 4 empty boxes + arrangements with 5 empty boxes)} \]

Substituting the values we calculated:

\[ = 5^5 - (5 + 0) = 5^5 - 5 \]

### Step 5: Calculate Final Value
Now we compute \(5^5\):

\[ 5^5 = 3125 \]

Thus, the final result is:

\[ 3125 - 5 = 3120 \]

### Final Answer
The number of ways in which we can put 5 different balls in 5 different boxes such that at most 3 boxes are empty is:

\[ \boxed{3120} \]