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 if all coins are identical and two pots are also identical?

Options

A. 2
B. 10
C. 9
D. 11

Answer

To solve the problem of distributing 4 identical coins into 3 pots where 2 pots are identical, we can follow these steps:

### Step 1: Understand the Problem
We have 4 identical coins and 3 pots (let's label them as A, B, and C). However, pots B and C are identical. This means that we need to consider the distributions that treat pots B and C as the same.

**Hint:** Identify the unique characteristics of the pots and coins in the problem.

### Step 2: Initial Distribution
Since we have 4 coins and 3 pots, we can start by placing one coin in each of the two identical pots (B and C). This will help us simplify the distribution.

**Hint:** Consider how many coins you can initially place in the pots to reduce the problem.

### Step 3: Remaining Coins
After placing one coin in each of the two identical pots (B and C), we have 4 - 2 = 2 coins left to distribute. Now, we need to distribute these 2 remaining coins into the three pots (A, B, and C).

**Hint:** Calculate how many coins are left after the initial distribution.

### Step 4: Distribution of Remaining Coins
Now we need to distribute the 2 remaining coins into the pots. Since pots B and C are identical, we will treat them as one pot for the purpose of distribution.

We can denote the number of coins in pot A as \(x\) and the number of coins in the combined pots B and C as \(y\). Therefore, we need to solve the equation:
\[ x + y = 2 \]

**Hint:** Set up an equation based on the remaining coins and the pots.

### Step 5: Finding Non-Negative Integer Solutions
The equation \(x + y = 2\) has the following non-negative integer solutions:
1. \(x = 2, y = 0\) (All coins in pot A)
2. \(x = 1, y = 1\) (One coin in pot A, one coin in the combined pots B and C)
3. \(x = 0, y = 2\) (No coins in pot A, two coins in the combined pots B and C)

Thus, the possible distributions are:
- (2, 0) - 2 coins in pot A, 0 in pots B and C
- (1, 1) - 1 coin in pot A, 1 in pots B and C
- (0, 2) - 0 coins in pot A, 2 in pots B and C

**Hint:** List all possible distributions based on the equation you set up.

### Step 6: Count the Unique Distributions
Since pots B and C are identical, the distributions (1, 1) and (0, 2) are counted as unique ways. Therefore, we have:
1. All coins in pot A (2, 0)
2. One coin in pot A and one coin in the identical pots (1, 1)
3. Two coins in the identical pots (0, 2)

Thus, the total number of ways to distribute the coins is **3**.

**Final Answer:** The number of ways to distribute the coins is **3**.