1. **Problem statement:** We have a group of students consisting of 3 freshmen, 10 sophomores, 5 juniors, and 6 seniors. We want to find the probability of choosing exactly 3 seniors when selecting 5 members at random for a committee. 2. **Total number of students:** $$3 + 10 + 5 + 6 = 24$$ 3. **Total ways to choose 5 members from 24:** We use combinations since order does not matter: $$\binom{24}{5}$$ 4. **Number of ways to choose exactly 3 seniors:** Choose 3 seniors from 6: $$\binom{6}{3}$$ 5. **Number of ways to choose the remaining 2 members from non-seniors:** Non-seniors count: $$3 + 10 + 5 = 18$$ Choose 2 from these 18: $$\binom{18}{2}$$ 6. **Number of favorable outcomes:** $$\binom{6}{3} \times \binom{18}{2}$$ 7. **Probability formula:** $$P(\text{exactly 3 seniors}) = \frac{\binom{6}{3} \times \binom{18}{2}}{\binom{24}{5}}$$ 8. **Calculate each combination:** $$\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$$ $$\binom{18}{2} = \frac{18 \times 17}{2 \times 1} = 153$$ $$\binom{24}{5} = \frac{24 \times 23 \times 22 \times 21 \times 20}{5 \times 4 \times 3 \times 2 \times 1} = 42504$$ 9. **Calculate numerator:** $$20 \times 153 = 3060$$ 10. **Calculate probability:** $$P = \frac{3060}{42504}$$ 11. **Simplify fraction:** Divide numerator and denominator by 12: $$\frac{\cancel{12}255}{\cancel{12}3542}$$ So, $$P = \frac{255}{3542} \approx 0.072$$ **Final answer:** The probability of choosing exactly 3 seniors is approximately **0.072** or **7.2%**.