A certain country has a cricket squad of 16 people, consisting of 7 batsmen, 5 bowlers, 2 all-rounders, and 2 wicket-keepers. The manager chooses a team of 11 players consisting of 5 batsmen, 4 bowlers, 1 all-rounder, and 1 wicket-keeper.
A team of 6 people is to be chosen from 5 swimmers, 7 athletes, and 4 cyclists. There must be at least 1 from each activity and there must be more athletes than cyclists. Find the number of different ways in which the team can be chosen.
A bunch of flowers consists of a mixture of roses, tulips and daffodils. Tom orders a bunch of 7 flowers from a shop to give to a friend. There must be at least 2 of each type of flower. The shop has 6 roses, 5 tulips and 4 daffodils, all different from each other. Find the number of different bunches of flowers that are possible.
Hannah chooses 5 singers from 15 applicants to appear in a concert. Of the 15 applicants, 10 are female and 5 are male. Hannah's friend Ami would like the group of 5 performers to include more males than females.
(iii) Find the number of different selections of 5 performers with more males than females.
(iv) Two of the applicants are Mr and Mrs Blake. Find the number of different selections that include Mr and Mrs Blake and also fulfil Amiโs requirement.
There are 7 Chinese, 6 European and 4 American students at an international conference. Four of the students are to be chosen to take part in a television broadcast. Find the number of different ways the students can be chosen if at least one Chinese and at least one European student are included.
A group of 8 friends travels to the airport in two taxis, P and Q. Each taxi can take 4 passengers.
The 8 friends divide themselves into two groups of 4, one group for taxi P and one group for taxi Q, with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done.
A committee of 6 people is to be chosen at random from 7 men and 9 women. Find the probability that there are no men on the committee.
Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks.
She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each type of ornament. How many different selections can Cherry make?
Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot both be in the group together.
(a) Find the number of ways in which a committee of 6 people can be chosen from 6 men and 8 women if it must include 3 men and 3 women.
A different committee of 6 people is to be chosen from 6 men and 8 women. Three of the 6 men are brothers.
(b) Find the number of ways in which this committee can be chosen if there are no restrictions on the numbers of men and women, but it must include no more than two of the brothers.
Sandra wishes to buy some applications (apps) for her smartphone but she only has enough money for 5 apps in total. There are 3 train apps, 6 social network apps and 14 games apps available. Sandra wants to have at least 1 of each type of app. Find the number of different possible selections of 5 apps that Sandra can choose.
Find the number of ways in which 9 different computer games can be shared out between Wainah, Jingyi, and Hebe so that each person receives an odd number of computer games.
The 50 members of a club include both the club president and the club treasurer. All 50 members want to go on a coach tour, but the coach only has room for 45 people. In how many ways can 45 members be chosen if both the club president and the club treasurer must be included?
A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done
A school club has members from 3 different year-groups: Year 1, Year 2 and Year 3. There are 7 members from Year 1, 2 members from Year 2 and 2 members from Year 3. Five members of the club are selected. Find the number of possible selections that include at least one member from each year-group.
A shop has 7 different mountain bicycles, 5 different racing bicycles and 8 different ordinary bicycles on display. A cycling club selects 6 of these 20 bicycles to buy.
How many different selections can be made if there must be no more than 3 mountain bicycles and no more than 2 of each of the other types of bicycle?
There are 10 spaniels, 14 retrievers and 6 poodles at a dog show. 7 dogs are selected to go through to the final.
How many selections of 7 different dogs can be made if there must be at least 1 spaniel, at least 2 retrievers and at least 3 poodles?
A town council plans to plant 12 trees along the centre of a main road. The council buys the trees from a garden centre which has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees for sale.
How many different selections of 12 trees can be made if there must be at least 2 of each type of tree?
Four families go to a theme park together. Mr and Mrs Lin take their 2 children. Mr OโConnor takes his 2 children. Mr and Mrs Ahmed take their 3 children. Mrs Burton takes her son.
Once inside the theme park, the children go on the roller-coaster. Each roller-coaster car holds 3 people.
In how many different ways can the 8 children be divided into two groups of 3 and one group of 2 to go on the roller-coaster?
A chess team of 2 girls and 2 boys is to be chosen from the 7 girls and 6 boys in the chess club. Find the number of ways this can be done if 2 of the girls are twins and are either both in the team or both not in the team.