1. **State the problem:** Simplify the expression $\frac{7!}{5!} \times 7!$. 2. **Recall the factorial definition:** For any positive integer $n$, $n! = n \times (n-1) \times \cdots \times 1$. 3. **Simplify the fraction $\frac{7!}{5!}$:** $$\frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!}$$ Cancel the common $5!$: $$= 7 \times 6$$ 4. **Calculate $7 \times 6$:** $$7 \times 6 = 42$$ 5. **Rewrite the original expression:** $$\frac{7!}{5!} \times 7! = 42 \times 7!$$ 6. **Calculate $7!$:** $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$$ 7. **Multiply $42 \times 5040$:** $$42 \times 5040 = 211680$$ **Final answer:** $$\boxed{211680}$$