1. **Stating the problem:** We need to solve the expression involving factorials: $\frac{11!}{3! \times 3! \times 2! \times 2!}$. 2. **Recall the factorial definition:** For any positive integer $n$, $n! = n \times (n-1) \times (n-2) \times \cdots \times 1$. 3. **Write the expression clearly:** $$\frac{11!}{3! \times 3! \times 2! \times 2!}$$ 4. **Calculate each factorial:** - $11! = 39916800$ - $3! = 6$ - $2! = 2$ 5. **Substitute the values:** $$\frac{39916800}{6 \times 6 \times 2 \times 2}$$ 6. **Multiply the denominator:** $$6 \times 6 = 36$$ $$2 \times 2 = 4$$ $$36 \times 4 = 144$$ 7. **Simplify the fraction:** $$\frac{39916800}{144}$$ 8. **Perform the division:** $$39916800 \div 144 = 277200$$ **Final answer:** $$\boxed{277200}$$