Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
In a group of 10 friends, how many different ways are there to:
a) sit 4 of them in a row
b) choose a group of 4
Solutions
Get Ready
Questions
How many ways are there of arranging the 4 letters A, A, B, C to give a different answer?
How many ways are there of arranging the 5 letters A, A, A, B, C to give a different answer?
Solutions
Notes
The total number of permutations of \(n\) objects, with \(k\) identical, is$$\frac{n!}{k!}$$
Examples and Your Turns
Example
How many ways are there of permuting the letters of the word CIRCLE?
Your Turn
How many ways are there of permuting the letters of the word MATHEMATICIAN?
Notes
Think about…
How many permutations of the word SQUARE have the Q and U next to each other?
If a group of objects have to be kept together then you have two approaches:
treat the objects that need to be kept together as a single “block”, and find the number of permutations of the blocks
remove the block, find the permutations of the rest and then count how many places it can be reinserted (choose from the gaps)
In both cases, you also need to calculate the number of permutation within the block.
Examples and Your Turns
Example
There are 7 books on a shelf. If I want to keep three of the books next to each other, how many ways are there for me to arrange the bookshelf?
Your Turn
10 students sit in a row. In how many ways can this happen if students A, B and C insist on sitting next to each other?
Your Turn
Jerry has a collection of 13 shirts. He wants to keep the 3 blue shirts next to each other, and the 4 white shirts next to each other. In how many ways can the shirts be arranged in his cupboard?
Your Turn
In how many ways can you arrange the letters of the word BINOMIAL if the letters B, N and M must be next to each other?
Notes
Think about…
How many permutations of the word SQUARE have none of the vowels together?
If \(k\) objects have to be kept apart, then we must follow these steps:
permute the other objects
permute the \(k\) objects
choose the gaps to place the \(k\) objects
Examples and Your Turns
Example
How many ways are there to permute the letters of MONDAY if the vowels must not be next to each other?
Your Turn
In a group of 10 students, Adam, Ben and Charlie must not be sat next to each other. How many ways are there to arrange the students in a row?
Your Turn
When designing the seating plan for his class of 15 students, Mr Ahmed wants to keep the four boys Aaron, Barack, Carlos and Declan separate from each other.
At the same time, he wants to sit the three girls Louisa, Moana and Noor together.
In how many ways can Mr Ahmed arrange his seating plan?
Your Turn
There are 12 seats in a classroom and 12 students in the class. Students A, B and C want to sit next to each other (they must be in the same row). In how many ways can this be done if:
a) there is 1 row of 12 seats?
b) there are 4 rows of 3 seats?
c) there are 3 rows of 4 seats?
d) there are 2 rows of 6 seats?
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.