Find the remainder in expression–
`(1! + 2! + 3! + …..10!)/(7)`

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the remainder of the expression \( \frac{1! + 2! + 3! + \ldots + 10!}{7} \), we can break it down into manageable steps.

### Step-by-Step Solution:

1. **Calculate Factorials**: First, we need to calculate the factorials from \( 1! \) to \( 6! \) since \( 7! \) and higher will be divisible by \( 7 \).

- \( 1! = 1 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 5! = 120 \)
- \( 6! = 720 \)

2. **Sum the Factorials**: Next, we sum these factorials.

\[ 1! + 2! + 3! + 4! + 5! + 6! = 1 + 2 + 6 + 24 + 120 + 720 \]

Calculating the sum:
\[ 1 + 2 = 3 \]
\[ 3 + 6 = 9 \]
\[ 9 + 24 = 33 \]
\[ 33 + 120 = 153 \]
\[ 153 + 720 = 873 \]

3. **Consider Higher Factorials**: Since \( 7! \), \( 8! \), \( 9! \), and \( 10! \) all contain \( 7 \) as a factor, they will contribute \( 0 \) to the remainder when divided by \( 7 \).

4. **Calculate the Remainder**: Now, we need to find the remainder when \( 873 \) is divided by \( 7 \).

\[ 873 \div 7 = 124 \quad \text{(integer part)} \]
\[ 124 \times 7 = 868 \]
\[ 873 - 868 = 5 \]

5. **Final Result**: The remainder when \( 1! + 2! + 3! + \ldots + 10! \) is divided by \( 7 \) is \( 5 \).

### Conclusion:
The remainder of the expression \( \frac{1! + 2! + 3! + \ldots + 10!}{7} \) is \( 5 \).

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