In how many ways we can put 5 letters into 5 corresponding envelopes so that atleast one letter go the wrong envelope?

Answer

To solve the problem of how many ways we can put 5 letters into 5 corresponding envelopes such that at least one letter goes into the wrong envelope, we can follow these steps:

### Step 1: Calculate the total number of arrangements
First, we need to find the total number of ways to arrange 5 letters into 5 envelopes. Since each letter has a unique envelope, this is a permutation problem where we have 5 distinct items.

The total number of arrangements can be calculated using the factorial of the number of items:
\[ \text{Total arrangements} = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 2: Identify the case where all letters are in the correct envelopes
Next, we need to consider the scenario where all letters are placed in their corresponding envelopes. There is only one way for this to happen, which is when each letter goes into its assigned envelope.

\[ \text{Ways with all letters correct} = 1 \]

### Step 3: Calculate the number of arrangements with at least one letter in the wrong envelope
To find the number of ways in which at least one letter is in the wrong envelope, we can subtract the number of arrangements where all letters are correct from the total arrangements.

\[ \text{Ways with at least one letter wrong} = \text{Total arrangements} - \text{Ways with all letters correct} \]
\[ = 120 - 1 = 119 \]

### Final Answer
Thus, the number of ways to put 5 letters into 5 corresponding envelopes such that at least one letter goes into the wrong envelope is:
\[ \boxed{119} \]