The number of ways in which 6 different balls can be put in two boxes of different sizes so that no box is empty is

Options

A. 64
B. 62
C. 60
D. 30

Answer

To solve the problem of distributing 6 different balls into 2 boxes of different sizes such that no box is empty, we can follow these steps:

### Step 1: Determine the total ways to distribute the balls
Each of the 6 balls can go into either of the 2 boxes. Therefore, for each ball, there are 2 choices (Box A or Box B).

The total number of ways to distribute the balls without any restrictions is given by:
\[ 2^6 \]

### Step 2: Calculate the total combinations
Calculating \(2^6\):
\[ 2^6 = 64 \]

### Step 3: Exclude the cases where at least one box is empty
However, we need to exclude the cases where at least one box is empty. There are 2 such cases:
1. All balls in Box A (Box B is empty).
2. All balls in Box B (Box A is empty).

### Step 4: Subtract the empty cases
So, we subtract these 2 cases from the total:
\[ 64 - 2 = 62 \]

### Conclusion
Thus, the number of ways to distribute 6 different balls into 2 boxes such that no box is empty is:
\[ \boxed{62} \]

---