A group consists of 4 girls and seven boys. In how many ways can a team of 5 members be selected if the
team has
(i) no girlThat means 'all boys'. 7 boys choose 5 = 7C5 = 21 ways
(ii) at least one boy and one girl.It's impossible not to have at least 1 boy.
So this is the number of ways to have any 5 of the 11 people
MINUS the cases where they're all boys, which is the result
of part (i)
So it's
(11 people choose 5) MINUS (7 boys choose 5) = 11C5 - 7C5 = 462-21 = 441
(iii) At least 3 girls.[Here is a case where you learn that AND implies MULTIPLICATION
and OR implies ADDITION.]
Case 1: 3 girls AND 2 boys
4 girls choose 3 AND 7 boys choose 2 = (4C3)(7C2) = (4)(21) = 84
[Notice that we MULTIPLIED due to AND]
OR
Case 2: 4 girls AND 1 boy
4 girls choose all 4 AND 7 boys choose 1 = 4C4+7C1 = (1)(7) = 7
[Notice that we MULTIPLIED due to AND]
Case 1 OR case 2 = 84+7 = 91
[Notice that we ADDED due to OR]
Edwin