The number of ways in which 13 gold coins can be distributed among three persons such that each one gets at least two gold coins is

Options

A. 36
B. 24
C. 42
D. 6

Answer

To solve the problem of distributing 13 gold coins among three persons such that each person gets at least two coins, we can follow these steps:

### Step-by-Step Solution:

1. **Initial Distribution**: Since each of the three persons must receive at least 2 coins, we start by giving 2 coins to each person. This means we distribute \(2 \times 3 = 6\) coins.

2. **Calculate Remaining Coins**: After giving out 6 coins, we have:
\[ 13 - 6 = 7 \text{ coins remaining} \]

3. **Distributing Remaining Coins**: Now, we need to distribute these 7 remaining coins among the 3 persons. There are no restrictions on how many coins a person can receive, meaning they can receive 0 or more coins.

4. **Using the Stars and Bars Theorem**: To find the number of ways to distribute \(n\) indistinguishable items (the remaining coins) into \(r\) distinguishable boxes (the persons), we can use the "stars and bars" theorem. The formula is given by:
\[ C(n + r - 1, r - 1) \]
where \(n\) is the number of indistinguishable items to distribute, and \(r\) is the number of distinguishable boxes.

5. **Applying the Formula**: In this case, \(n = 7\) (remaining coins) and \(r = 3\) (persons). Thus, we need to calculate:
\[ C(7 + 3 - 1, 3 - 1) = C(9, 2) \]

6. **Calculating the Combination**: The combination \(C(9, 2)\) is calculated as follows:
\[ C(9, 2) = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

7. **Final Answer**: Therefore, the number of ways to distribute the 13 gold coins among 3 persons such that each one gets at least 2 coins is:
\[ \text{Answer} = 36 \]