Evaluate the factorial expression. \( \frac{107!}{105!} \) \( \frac{107!}{105!}=\square \)

We start with the expression: \[ \frac{107!}{105!} \] Recall that the factorial function is defined by: \[ n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \] So, we write: \[ 107! = 107 \times 106 \times 105! \] Substitute this into the original expression: \[ \frac{107!}{105!} = \frac{107 \times 106 \times 105!}{105!} \] Cancel out the \(105!\) terms: \[ \frac{107 \times 106 \times \cancel{105!}}{\cancel{105!}} = 107 \times 106 \] Now, calculate the product: \[ 107 \times 106 = 107 \times (100 + 6) = 10700 + 642 = 11342 \] Thus, the final answer is: \[ \frac{107!}{105!} = 11342 \]