In how many ways can 3 letters be posted in four letter boxes in a village ? If all the three letters are not posted in the same letter box. Find the corresponding number of ways of posting.

Answer

To solve the problem of how many ways 3 letters can be posted in 4 letter boxes such that not all letters are posted in the same box, we can follow these steps:

### Step 1: Calculate Total Ways Without Restrictions
Each letter can be posted in any of the 4 boxes. Therefore, for each of the 3 letters, we have 4 choices.

- For the first letter, there are 4 choices.
- For the second letter, there are also 4 choices.
- For the third letter, there are again 4 choices.

Thus, the total number of ways to post the letters without any restrictions is:
\[ 4 \times 4 \times 4 = 4^3 = 64 \text{ ways} \]

### Step 2: Exclude the Case Where All Letters Are in the Same Box
Now, we need to exclude the cases where all 3 letters are posted in the same box. Since there are 4 boxes, there are exactly 4 ways to post all letters in the same box (one for each box).

### Step 3: Calculate Valid Ways
To find the valid ways of posting the letters such that not all letters are in the same box, we subtract the 4 cases (where all letters are in the same box) from the total ways calculated in Step 1.

\[ \text{Valid ways} = \text{Total ways} - \text{Ways where all letters are in the same box} \]
\[ \text{Valid ways} = 64 - 4 = 60 \]

### Final Answer
Therefore, the total number of ways to post 3 letters in 4 letter boxes such that not all letters are posted in the same box is:
\[ \boxed{60} \]