1. Binomial Expansion 2
L.O.
All pupils can follow where the binomial expansion comes from
All pupils understand binomial notation
All pupils can use the binomial theorem to complete expansions

2. 1)Three people were in a race – ABC How may different options of
Main 1:
where the binomial expansion comes from
1)Three people were in a race – ABC
How may different options of
finishing positions are there?
2)Two out of those 3 people were
going to be picked- the order they are
picked doesn’t matter.
How many outcomes are there for this?

3. 1)Three people were in a race – ABC How may different options of
Main 1:
where the binomial expansion comes from
1)Three people were in a race – ABC
How may different options of
finishing positions are there?
You may have worked the following options out:
ABC BAC CAB ACB BCA CBA

4. 1)Three people were in a race – ABC How may different options of
Main 1:
where the binomial expansion comes from
1)Three people were in a race – ABC
How may different options of
finishing positions are there?
You may have worked the following options out:
ABC BAC CAB ACB BCA CBA
So 6 is the answer. However there is a more mathematical method which becomes more helpful with more options
eg. in a race of 20 people

5. 1st 2nd 3rd Main 1: 1)Three people were in a race – ABC
where the binomial expansion comes from
1)Three people were in a race – ABC
How may different options of
finishing positions are there?
You may have worked the following options out:
ABC BAC CAB ACB BCA CBA
So 6 is the answer. However there is a more mathematical method which becomes more helpful with more options
eg. in a race of 20 people
1st
2nd
3rd

6. 1st 2nd 3rd Main 1: 1)Three people were in a race – ABC
where the binomial expansion comes from
1)Three people were in a race – ABC
How may different options of
finishing positions are there?
You may have worked the following options out:
ABC BAC CAB ACB BCA CBA
So 6 is the answer. However there is a more mathematical method which becomes more helpful with more options
eg. in a race of 20 people
1st
2nd
3rd
3 options for people who could come first

7. 1st 2nd 3rd Main 1: 1)Three people were in a race – ABC
where the binomial expansion comes from
1)Three people were in a race – ABC
How may different options of
finishing positions are there?
You may have worked the following options out:
ABC BAC CAB ACB BCA CBA
So 6 is the answer. However there is a more mathematical method which becomes more helpful with more options
eg. in a race of 20 people
1st
2nd
3rd
So only 1 person left for 3rd
3 options for people who could come first
Once 1st is decided only 2 people left for 2nd

8. 1st 2nd 3rd Main 1: 1)Three people were in a race – ABC
where the binomial expansion comes from
1)Three people were in a race – ABC
How may different options of
finishing positions are there?
You may have worked the following options out:
ABC BAC CAB ACB BCA CBA
So 6 is the answer. However there is a more mathematical method which becomes more helpful with more options
eg. in a race of 20 people
1st
2nd
3rd
3 options for people who could come first
Once 1st is decided only 2 people left for 2nd

9. Main 1:
where the binomial expansion comes from
1st
2nd
3rd
3 options of people who could come first
So only 1 person left for 3rd
Once 1st is decided only 2 people left for 2nd
So x 2 x 1 = 6

10. Main 1:
where the binomial expansion comes from
1st
2nd
3rd
3 options of people who could come first
So only 1 person left for 3rd
Once 1st is decided only 2 people left for 2nd
So x 2 x 1 = 6
This can be written as 3!  We say ‘three factorial’
NOTICE- ORDER MATTERS  ABC is Different to CBA

11. Main 1:
where the binomial expansion comes from
1st
2nd
3rd
3 options of people who could come first
So only 1 person left for 3rd
Once 1st is decided only 2 people left for 2nd
So x 2 x 1 = 6
This can be written as 3!  We say ‘three factorial’
NOTICE- ORDER MATTERS  ABC is Different to CBA
So in a race of 20 people the number of different outcomes of finishing positions would have 20 options for 1st place, 19 left for second, 18 left for third etc so would be TWENTY FACTORIAL- 20!

12. where the binomial expansion comes from
Main 1:
where the binomial expansion comes from
FACTORIALS
Work out these : 5! 7! 8! 4!
d) 10! 6!

13. where the binomial expansion comes from
Main 1:
where the binomial expansion comes from
FACTORIALS
Answers to the :
5! 5x4x3x2x1 =120
7! 7x6x5x4x3x2x1= 5040
8! 8x7x6x5x4x3x2x1 =1680
4! x3x2x1
d) 10! 10x9x8x7x6x5x4x3x2x1 = 5040
6! 6x5x4x3x2x1

14. Main 1: 2)Two out of those 3 people were
where the binomial expansion comes from
2)Two out of those 3 people were
going to be picked- the order they are
picked doesn’t matter.
How many outcomes are there for this?
You may have worked the following options out:
AB AC BC
Same as BA Same as CA Same as CB

15. Main 1: 2)Two out of those 3 people were
where the binomial expansion comes from
2)Two out of those 3 people were
going to be picked- the order they are
picked doesn’t matter.
How many outcomes are there for this?
You may have worked the following options out:
AB AC BC
Same as BA Same as CA Same as CB
So 3 is the answer. This is a tiny bit harder to work out mathematically:

16. 1st 2nd Main 1: 2)Two out of those 3 people were
where the binomial expansion comes from
2)Two out of those 3 people were
going to be picked- the order they are
picked doesn’t matter.
How many outcomes are there for this?
You may have worked the following options out:
AB AC BC
Same as BA Same as CA Same as CB
So 3 is the answer. This is a tiny bit harder to work out mathematically:
1st
2nd
There are 3 choices for the first pick

17. 1st 2nd Main 1: 2)Two out of those 3 people were
where the binomial expansion comes from
2)Two out of those 3 people were
going to be picked- the order they are
picked doesn’t matter.
How many outcomes are there for this?
You may have worked the following options out:
AB AC BC
Same as BA Same as CA Same as CB
So 3 is the answer. This is a tiny bit harder to work out mathematically:
1st
2nd
There are 3 choices for the first pick
So that leaves 2 choices for the second pick

18. 1st 2nd Main 1: 2)Two out of those 3 people were
where the binomial expansion comes from
2)Two out of those 3 people were
going to be picked- the order they are
picked doesn’t matter.
How many outcomes are there for this?
You may have worked the following options out:
AB AC BC
Same as BA Same as CA Same as CB
So 3 is the answer. This is a tiny bit harder to work out mathematically:
1st
2nd
There are 3 choices for the first pick
So that leaves 2 choices for the second pick
So 3 x 2 = but that gives the options if AB is the same as BA,
AC as CA and BC as CB
But the question states the order doesn’t matter- so 3x2 =3 2

19. Main 1:
where the binomial expansion comes from
3 out of those 5 people were going to be picked- the order they are picked doesn’t matter.
How many outcomes are there for this?
Outcomes:
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
So 10 is the answer, but mathematically:

20. Main 1:
where the binomial expansion comes from
3 out of those 5 people were going to be picked- the order they are picked doesn’t matter.
How many outcomes are there for this?
Outcomes:
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
So 10 is the answer, but mathematically:
Take ABC as an example if the order mattered these are all the options: ABC ACB
BAC BCA
CAB CBA

21. Main 1:
where the binomial expansion comes from
3 out of those 5 people were going to be picked- the order they are picked doesn’t matter.
How many outcomes are there for this?
Outcomes:
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
So 10 is the answer, but mathematically:
Take ABC as an example if the order mattered these are all the options: ABC ACB
BAC BCA
CAB CBA
1st
2nd
3nd
There are 3 choices
for the third pick
There are 5 choices
for the first pick
So that leaves 4 choices
for the second pick

22. Main 1:
where the binomial expansion comes from
3 out of those 5 people were going to be picked- the order they are picked doesn’t matter.
How many outcomes are there for this?
Outcomes:
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
So 10 is the answer, but mathematically:
Take ABC as an example if the order mattered these are all the options: ABC ACB
BAC BCA
CAB CBA
1st
2nd
3nd
There are 3 choices
for the third pick
There are 5 choices
for the first pick
So that leaves 4 choices
for the second pick
This is like the first question - 3 choices put in order which is 3!

23. Main 1:
where the binomial expansion comes from
3 out of those 5 people were going to be picked- the order they are picked doesn’t matter.
How many outcomes are there for this?
Outcomes:
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
So 10 is the answer, but mathematically:
Take ABC as an example if the order mattered these are all the options: ABC ACB
BAC BCA
CAB CBA
1st
2nd
3nd
There are 3 choices
for the third pick
There are 5 choices
for the first pick
So that leaves 4 choices
for the second pick
This is like the first question - 3 choices put in order which is 3!
So 5 x 4 x 3 =60
But the question states the order doesn’t matter- so 5x4x3 = 5x4x3 =10
3! x2

24. where the binomial expansion comes from
Main 1:
where the binomial expansion comes from
5 options 3 Choices
So 5x4x3 = 5x4x3 =10
3! x2
3 options 2 Choices
So 3x2 =3 2
Can you see any rule?

25. where the binomial expansion comes from
Main 1:
where the binomial expansion comes from
5 options 3 Choices
So 5x4x3 = 5x4x3 =10
3! x2
3 options 2 Choices
So 3x2 =3 2

26. where the binomial expansion comes from
Main 1:
where the binomial expansion comes from
5 options 3 Choices
So 5x4x3 = 5x4x3 =10
3! x2
3 options 2 Choices
So 3x2 =3 2
COMBINATION RULE:
n options r choices n!
(n-r)!r!

27. Binomial Expansion 2
L.O.
All pupils can follow where the binomial expansion comes from
All pupils understand binomial notation
All pupils can use the binomial theorem to complete expansions

28. Main 2: (n-r)!r! n options r choices binomial notation
5 options 3 Choices
So 5x4x3 = 5x4x3 =10
3! x2
3 options 2 Choices
So 3x2 =3 2
COMBINATION RULE:
n options r choices n!
(n-r)!r!
5 options 3 Choices
So 5x4x3x2x1
2x1x3x2x1

29. Main 2: (n-r)!r! n options r choices binomial notation
5 options 3 Choices
So 5x4x3 = 5x4x3 =10
3! x2
3 options 2 Choices
So 3x2 =3 2
COMBINATION RULE:
n options r choices n!
(n-r)!r!
5 options 3 Choices
So 5x4x3x2x1
2x1x3x2x1

30. Main 2: (n-r)!r! n options r choices binomial notation
5 options 3 Choices
So 5x4x3 = 5x4x3 =10
3! x2
3 options 2 Choices
So 3x2 =3 2
COMBINATION RULE:
n options r choices n!
(n-r)!r!
5 options 3 Choices
So 5x4x3x2x1
2x1x3x2x1
3 options 2 Choices
So 3x2x1
1x2x1

31. Main 2: (n-r)!r! n options r choices binomial notation
5 options 3 Choices
So 5x4x3 = 5x4x3 =10
3! x2
3 options 2 Choices
So 3x2 =3 2
COMBINATION RULE:
n options r choices n!
(n-r)!r!
5 options 3 Choices
So 5x4x3x2x1
2x1x3x2x1
3 options 2 Choices
So 3x2x1
1x2x1

32. Combinations can use 2 notations- you need to recognise both!
Main 2:
binomial notation
Combinations can use 2 notations- you need to recognise both! n r
ncr
n – is the total number of options
r - is the number of choices you will make
Says ‘ n choice r’

33. Combinations can use 2 notations- you need to recognise both!
Main 2:
use the binomial theorem to complete expansions
Combinations can use 2 notations- you need to recognise both!
ncr n r
n – is the total number of options
r - is the number of choices you will make
Says ‘ n choice r’
COMBINATION RULE:
n options r choices n!
(n-r)!r!

34. Binomial Expansion 2
L.O.
All pupils can follow where the binomial expansion comes from
All pupils understand binomial notation
All pupils can use the binomial theorem to complete expansions

35. How to use this for binomial expansion
Main 3:
use the binomial theorem to complete expansions
How to use this for binomial expansion
Eg. (1+x)4
ncr
Power of 4 so the coefficients will be:
4c c1 13 x c2 12x c3 11x c4 x4

36. How to use this for binomial expansion
Main 3:
use the binomial theorem to complete expansions
How to use this for binomial expansion
Eg. (1+x)4
ncr
Power of 4 so the coefficients will be:
4c c1 13 x c2 12x c3 11x c4 x4
Notice:
The powers of the two numbers add up to = n

37. How to use this for binomial expansion
Main 3:
use the binomial theorem to complete expansions
How to use this for binomial expansion
Eg. (1+x)4
ncr
Power of 4 so the coefficients will be:
4c c1 13 x c2 12x c3 11x c4 x4
Notice:
The powers of the two numbers add up to = n
The value of r is the same as the power of the second half of the term

38. How to use this for binomial expansion
Main 3:
use the binomial theorem to complete expansions
How to use this for binomial expansion
(2x-5)4
ncr
Power of 4 so the coefficients will be:
4c c c c c4

39. How to use this for binomial expansion
Main 3:
use the binomial theorem to complete expansions
How to use this for binomial expansion
(2x-5)4
ncr
Power of 4 so the coefficients will be:
4c c c c c4
Terms will be
(2x)4, (2x)3(-5), (2x)2(-5)2, (2x)(-5)3 ,(-5)4
16x4, -40x3 , 100x , x , 625

40. How to use this for binomial expansion
Main 3:
use the binomial theorem to complete expansions
How to use this for binomial expansion
(2x-5)4
ncr
Power of 4 so the coefficients will be:
4c c c c c4
Terms will be:
(2x)4, (2x)3(-5), (2x)2(-5)2, (2x)(-5)3 ,(-5)4
16x4, -40x3 , 100x , x , 625
Put them together
16x x x x

41. Binomial Expansion 2
L.O.
All pupils can follow where the binomial expansion comes from
All pupils understand binomial notation
All pupils can use the binomial theorem to complete expansions