Question

a group consist of 4 girls and 7 boys in how many ways can a team of 5 members be selected if the team has 1) no girl 2)atlest one boy and one girl 3)at least 3 girls

Answer 1

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 girl That 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]