There are 6 letters and 3 post boxes. The number of ways in which these letters can be posted is -

Answer

To solve the problem of distributing 6 letters into 3 post boxes, we can use the concept of combinations and the principle of distribution. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have 6 letters and 3 post boxes. We need to find out how many ways we can distribute these letters into the boxes. Each box must contain at least one letter.

### Step 2: Set Up the Equation
Let’s denote the number of letters in each box as \( A \), \( B \), and \( C \). The equation we need to satisfy is:
\[ A + B + C = 6 \]
where \( A, B, C \geq 1 \) (since each box must have at least one letter).

### Step 3: Adjust for Minimum Letters
To simplify the equation, we can redefine the variables. Let:
\[ A' = A - 1 \]
\[ B' = B - 1 \]
\[ C' = C - 1 \]
Now, \( A', B', C' \geq 0 \). The equation becomes:
\[ A' + B' + C' = 3 \]

### Step 4: Use Stars and Bars Theorem
We can use the stars and bars theorem to find the number of non-negative integer solutions to the equation \( A' + B' + C' = 3 \). The formula for the number of solutions is given by:
\[ \text{Number of solutions} = \binom{n + k - 1}{k - 1} \]
where \( n \) is the total number of items to distribute (3 in this case), and \( k \) is the number of boxes (3 here).

Thus, we have:
\[ \text{Number of solutions} = \binom{3 + 3 - 1}{3 - 1} = \binom{5}{2} = 10 \]

### Step 5: Calculate the Total Distributions
Now, for each distribution of letters into the boxes, we need to consider the arrangements of the letters themselves. Since the letters are distinct, the total number of ways to arrange 6 letters is \( 6! \).

### Step 6: Combine the Results
The total number of ways to distribute the letters into the boxes is:
\[ \text{Total Ways} = \text{Number of distributions} \times \text{Arrangements of letters} \]
\[ \text{Total Ways} = 10 \times 6! = 10 \times 720 = 7200 \]

### Final Answer
Thus, the total number of ways in which these letters can be posted is **7200**.

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