Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls
- if there are more girls than boys in the team,
- if three of the boys are cousins and are either all in the team or all not in the team.
Solution
(i) To have more girls than boys in the team of 5, the possible combinations are:
- 5 girls and 0 boys: \(\binom{8}{5} \times \binom{6}{0} = 56\)
- 4 girls and 1 boy: \(\binom{8}{4} \times \binom{6}{1} = 420\)
- 3 girls and 2 boys: \(\binom{8}{3} \times \binom{6}{2} = 840\)
\(Total = 56 + 420 + 840 = 1316\)
(ii) If three of the boys are cousins and are either all in the team or all not in the team, consider two cases:
- All cousins in the team: Choose 2 more from the remaining 11 (3 boys + 8 girls): \(\binom{11}{2} = 55\)
- All cousins not in the team: Choose 5 from the remaining 11 (3 boys + 8 girls): \(\binom{11}{5} = 462\)
\(Total = 55 + 462 = 517\)
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