Find the unit digit in `1!+2!+3!+4!+……….+50!`.

Options

A. 1
B. 3
C. 5
D. 8

Answer

To find the unit digit in the sum \(1! + 2! + 3! + 4! + \ldots + 50!\), we can follow these steps:

### Step 1: Calculate the factorials from \(1!\) to \(4!\)

1. **Calculate \(1!\)**:
\[ 1! = 1 \]
**Unit digit: 1**

2. **Calculate \(2!\)**:
\[ 2! = 2 \times 1 = 2 \]
**Unit digit: 2**

3. **Calculate \(3!\)**:
\[ 3! = 3 \times 2 \times 1 = 6 \]
**Unit digit: 6**

4. **Calculate \(4!\)**:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
**Unit digit: 4**

### Step 2: Identify the unit digits of \(5!\) and higher

5. **Calculate \(5!\)**:
\[ 5! = 5 \times 4! = 5 \times 24 = 120 \]
**Unit digit: 0**

From \(5!\) onwards, all factorials will end in 0 because they include the multiplication by 5 and at least one even number (2), which results in a factor of 10.

### Step 3: Sum the unit digits of \(1!\) to \(4!\)

6. **Sum the unit digits**:
\[ \text{Unit digit of } (1! + 2! + 3! + 4!) = 1 + 2 + 6 + 4 \]
\[ = 13 \]
The unit digit of 13 is **3**.

### Step 4: Conclusion

Since all factorials from \(5!\) to \(50!\) contribute a unit digit of 0, they do not affect the final unit digit of the sum.

Thus, the unit digit in \(1! + 2! + 3! + 4! + \ldots + 50!\) is **3**.