Three school prefects are to be chosen from four girls and five boys. What is the probability that : (a) only boys will be chosen ; (b) more girls than boys will be chosen ?
Explanation
4 girls and 5 boys (a) p(only boys) = \(\frac{^{5}C_{3}}{^{9}C_{3}}\) = \(\frac{5!}{2!3!} \times \frac{3!6!}{9!}\) = \(\frac{10 \times 1}{84}\) = \(\frac{5}{42}\) (b) p(more girls than boys) = p(3 girls and no boys) or p(2 girls and 1 boy) Number of ways of selecting = \(^{4}C_{3} \times ^{5}C_{0} + ^{4}C_{2} \times ^{5}C_{1}\) = \(4 \times 1 + 6 \times 5\) = 34 ways. Without restrictions, selection can be made in \(^{9}C_{3}\) ways. = \(\frac{9!}{6! 3!}\) = 84 ways. \(\therefore\) p(more girls than boys) = \(\frac{34}{84}\) = \(\frac{17}{42}\).