1. **Stating the problem:** We need to select a head boy, two deputy head boys, a head girl, and three deputy head girls from a student council of 14 girls and 16 boys. 2. **Understanding the selection:** - Head boy: choose 1 from 16 boys. - Deputy head boys: choose 2 from remaining 15 boys (since one is head boy). - Head girl: choose 1 from 14 girls. - Deputy head girls: choose 3 from remaining 13 girls (since one is head girl). 3. **Formulas used:** - Number of ways to choose $k$ items from $n$ is given by combinations: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ - For ordered positions like head boy, since only one is chosen, it's simply $16$ ways. 4. **Calculations:** - Head boy: $16$ ways. - Deputy head boys: $$\binom{15}{2} = \frac{15 \times 14}{2 \times 1} = 105$$ ways. - Head girl: $14$ ways. - Deputy head girls: $$\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286$$ ways. 5. **Total number of ways:** $$16 \times 105 \times 14 \times 286$$ 6. **Multiplying step-by-step:** - $16 \times 105 = 1680$ - $1680 \times 14 = 23520$ - $23520 \times 286 = 6726720$ 7. **Final answer:** There are $6726720$ ways to choose the positions. Hence, the correct choice is d. 6726720.