There are 3 pots and 4 coins. All these coins are to be distributed into these pots where any pot can contain any number of coins.
In how many ways all these coins can be distributed such that no pot is empty if all coins are identical but all pots are different?

Options

A. 6
B. 3
C. 9
D. 27

Answer

To solve the problem of distributing 4 identical coins into 3 different pots such that no pot is empty, we can follow these steps:

### Step 1: Ensure Each Pot Gets One Coin
Since we have 3 pots and we need to ensure that no pot is empty, we start by placing 1 coin in each of the 3 pots. This ensures that all pots have at least one coin.

- **Coins in pots after this step**:
- Pot A: 1 coin
- Pot B: 1 coin
- Pot C: 1 coin

### Step 2: Calculate Remaining Coins
After placing 1 coin in each of the 3 pots, we have used 3 coins. Since we started with 4 coins, we now have:

- **Remaining coins**: 4 - 3 = 1 coin

### Step 3: Distribute the Remaining Coin
Now, we have 1 identical coin left to distribute. This coin can go into any of the 3 pots (A, B, or C).

- **Ways to distribute the remaining coin**:
- It can go into Pot A
- It can go into Pot B
- It can go into Pot C

Thus, there are 3 different ways to distribute this remaining coin.

### Step 4: Conclusion
Therefore, the total number of ways to distribute the 4 identical coins into 3 different pots such that no pot is empty is **3**.

**Final Answer**: 3 ways

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