Find the unit place digit in the expression given below:
1! +2! + 3! + 4!+………..+ 20!

Options

A. 5
B. 0
C. 3
D. 9

Answer

To find the unit place digit in the expression \(1! + 2! + 3! + 4! + \ldots + 20!\), we can follow these steps:

### Step 1: Calculate the factorials from \(1!\) to \(4!\)
- \(1! = 1\)
- \(2! = 2\)
- \(3! = 6\)
- \(4! = 24\)

### Step 2: Find the unit place digits of these factorials
- The unit place digit of \(1!\) is \(1\).
- The unit place digit of \(2!\) is \(2\).
- The unit place digit of \(3!\) is \(6\).
- The unit place digit of \(4!\) is \(4\) (from \(24\)).

### Step 3: Calculate the sum of the unit place digits from \(1!\) to \(4!\)
Now, we sum these unit place digits:
\[ 1 + 2 + 6 + 4 = 13 \]
The unit place digit of \(13\) is \(3\).

### Step 4: Analyze factorials from \(5!\) to \(20!\)
For \(n \geq 5\), the factorial \(n!\) will always contain both \(2\) and \(5\) as factors, which means:
- \(5! = 120\) (unit digit is \(0\))
- \(6! = 720\) (unit digit is \(0\))
- \(7! = 5040\) (unit digit is \(0\))
- ...
- \(20!\) will also have a unit digit of \(0\).

### Step 5: Conclusion
Since all factorials from \(5!\) to \(20!\) contribute a unit digit of \(0\), they do not affect the unit place digit of the entire sum.

Thus, the unit place digit of the entire expression \(1! + 2! + 3! + 4! + \ldots + 20!\) is simply the unit place digit of \(1 + 2 + 6 + 4\), which is \(3\).

### Final Answer
The unit place digit in the expression \(1! + 2! + 3! + 4! + \ldots + 20!\) is \(3\).

---