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

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?

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

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

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

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

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

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

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 3 x 2 x 1 = 6

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 3 x 2 x 1 = 6 This can be written as 3!  We say ‘three factorial’ NOTICE- ORDER MATTERS  ABC is Different to CBA

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 3 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!

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!

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! 4x3x2x1 d) 10! 10x9x8x7x6x5x4x3x2x1 = 5040 6! 6x5x4x3x2x1

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

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 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

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

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 =6 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

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:

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

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

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!

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! 3x2

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

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

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

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

Main 2: (n-r)!r! n options r choices binomial notation 5 options 3 Choices So 5x4x3 = 5x4x3 =10 3! 3x2 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

Main 2: (n-r)!r! n options r choices binomial notation 5 options 3 Choices So 5x4x3 = 5x4x3 =10 3! 3x2 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

Main 2: (n-r)!r! n options r choices binomial notation 5 options 3 Choices So 5x4x3 = 5x4x3 =10 3! 3x2 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

Main 2: (n-r)!r! n options r choices binomial notation 5 options 3 Choices So 5x4x3 = 5x4x3 =10 3! 3x2 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

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’

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!

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

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: 4c0 14 4c1 13 x1 4c2 12x2 4c3 11x3 4c4 x4

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: 4c0 14 4c1 13 x1 4c2 12x2 4c3 11x3 4c4 x4 Notice: The powers of the two numbers add up to = n

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: 4c0 14 4c1 13 x1 4c2 12x2 4c3 11x3 4c4 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

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: 4c0 4c1 4c2 4c3 4c4 1 4 6 4 1

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: 4c0 4c1 4c2 4c3 4c4 1 4 6 4 1 Terms will be (2x)4, (2x)3(-5), (2x)2(-5)2, (2x)(-5)3 ,(-5)4 16x4, -40x3 , 100x2 , -250x , 625

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: 4c0 4c1 4c2 4c3 4c4 1 4 6 4 1 Terms will be: (2x)4, (2x)3(-5), (2x)2(-5)2, (2x)(-5)3 ,(-5)4 16x4, -40x3 , 100x2 , -250x , 625 Put them together 1 4 6 4 1 16x4 - 160x3 + 600x2 - 1000x + 625

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