Unit 1 - Number and Algebra
Get Ready
Questions
From a group of \(11\) students, \(3\) are to be chosen to meet with the principal. In how many different ways can the \(3\) students be chosen?
Solutions
Notes
The number of ways of choosing \(r\) objects from a group of \(n\) objects is given by$$\binom{n}{r}={}^{n} \mathrm{ C }_{r}=\frac{n!}{r!\left(n-r\right)!}$$This formula comes from the formula for permutations \({}^{n} \mathrm{ P }_{r}=\frac{n!}{\left(n-r\right)!}\), divided through by the number of way of arranging the \(r\) objects, which is \(r!\).
Note that we choose when the order does not matter.
Examples and Your Turns
Example
A group of 12 friends want to form a team for a 5-a-side football tournament.
(a) In how many different ways can a team of five be chosen?
(b) Rob and Amir are the only goalkeepers and they cannot play in other positions. If the team must include exactly one goalkeeper, in how many ways can the team be chosen?
Your Turn
How many different teams of 4 can be selected from a squad of 7 if:
(a) there are no restrictions?
(b) the team must include the captain?
Your Turn
A committee of 4 is to be chosen from a group of 7 year 13 students and 6 year 12 students. How many different committees can be chosen if:
(a) there are no restrictions?
(b) there must be exactly 2 from year 13 and 2 from year 12?
(c) at least one student must be chosen from each year group?
Your Turn
In a supermarket meal deal you can choose 5 items from a selection of 8 starters, 4 main courses and 7 desserts. How many different meal deals can be chosen if:
(a) there are no restrictions?
(b) there must be at least one of each course?
(c) there can be a maximum of 2 main courses?
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.