Committee Formation 0C7C02

1. **State the problem:** We have $14$ students and want to form a committee of $4$ people: $1$ chairperson, $1$ vice-chairperson, and $2$ members. 2. **Understand the roles:** The chairperson and vice-chairperson are distinct positions, so order matters for these two roles. The $2$ members are indistinguishable in terms of position, so order does not matter for them. 3. **Step 1: Choose the chairperson.** There are $14$ choices. 4. **Step 2: Choose the vice-chairperson.** After choosing the chairperson, $13$ students remain, so $13$ choices. 5. **Step 3: Choose the $2$ members.** After choosing chairperson and vice-chairperson, $12$ students remain. We choose $2$ members without regard to order, so number of ways is $$\binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66.$$ 6. **Calculate total number of committees:** Multiply the number of ways for each step: $$14 \times 13 \times 66 = 14 \times 13 \times 66.$$ 7. **Calculate intermediate multiplication:** $$14 \times 13 = 182,$$ so total is $$182 \times 66.$$ 8. **Calculate final multiplication:** $$182 \times 66 = 182 \times (60 + 6) = 182 \times 60 + 182 \times 6 = 10920 + 1092 = 12012.$$ 9. **Answer:** The number of different committees is $12012$, which corresponds to option C. **Final answer:** $12012$ (Option C).