Find the remainder in expression–
`(1! + 2! + 3! + ……. 100 !)/(5)`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the remainder of the expression \( \frac{1! + 2! + 3! + \ldots + 100!}{5} \), we will calculate the factorials from \( 1! \) to \( 100! \) and find the remainder of each factorial when divided by \( 5 \).

### Step-by-Step Solution:

1. **Calculate Factorials and Their Remainders:**
- \( 1! = 1 \)
- Remainder when \( 1 \) is divided by \( 5 \) is \( 1 \).
- \( 2! = 2 \)
- Remainder when \( 2 \) is divided by \( 5 \) is \( 2 \).
- \( 3! = 6 \)
- Remainder when \( 6 \) is divided by \( 5 \) is \( 1 \) (since \( 6 - 5 = 1 \)).
- \( 4! = 24 \)
- Remainder when \( 24 \) is divided by \( 5 \) is \( 4 \) (since \( 24 - 20 = 4 \)).
- \( 5! = 120 \)
- Remainder when \( 120 \) is divided by \( 5 \) is \( 0 \) (since \( 120 \) is divisible by \( 5 \)).
- For \( n! \) where \( n \geq 5 \), \( n! \) will also be divisible by \( 5 \) (since \( 5! \) and higher factorials include \( 5 \) as a factor). Thus, the remainder will be \( 0 \) for all \( n! \) where \( n \geq 5 \).

2. **Sum the Remainders:**
- From the calculations above, we find the remainders:
- \( 1! \equiv 1 \)
- \( 2! \equiv 2 \)
- \( 3! \equiv 1 \)
- \( 4! \equiv 4 \)
- \( 5! \equiv 0 \)
- \( 6! \equiv 0 \)
- ...
- \( 100! \equiv 0 \)
- Therefore, the total sum of the remainders is:
\[ 1 + 2 + 1 + 4 + 0 + 0 + \ldots + 0 = 1 + 2 + 1 + 4 = 8 \]

3. **Find the Final Remainder:**
- Now, we find the remainder of \( 8 \) when divided by \( 5 \):
\[ 8 \div 5 = 1 \quad \text{(quotient)} \quad \text{with a remainder of } 3. \]

Thus, the remainder when \( 1! + 2! + 3! + \ldots + 100! \) is divided by \( 5 \) is \( \boxed{3} \).