1. Binomial distribution
Combnations

2. Binomial distribution
KUS objectives
BAT use factorial notation to find the number of arrangements of some objects
Starter: Discuss how many ways are there:
Of choosing a Captain and deputy captain from 11 players
Of getting 3 heads and two tails when you flip a coin 5 times
Of winning the main UK lottery?
How sure are you of your answers?

3. This pattern is known as
Notes2
Consider the number of combinations from tossing coins:
1 coin
Outcome
No. of combinations
1 pick
This pattern is known as
Pascal’s Triangle
2 coins
Outcome
No. of combinations
2 picks
3 coins
Outcome
No. of combinations
3 picks
4 coins
Outcome
No. of combinations
4 picks
5 coins
Outcome
No. of combinations
5 picks
It gives the number of combinations that give r matches from n picks between 2 options (in this case heads or tails)
Eg how many ways can you get 3 tails out of 4 tosses?
4 ways

4. Eg how many ways can you get 3 heads out of 5 tosses?
Notes 3
This calculation is known as
Pascal realised that the number of combinations could be directly calculated from n and r: or
Number of combinations
and has a calculator key
Eg how many ways can you get 3 heads out of 5 tosses?
Possible combinations:
HHHTT
HHTTH
HTTHH
TTHHH
HHTHT
HTHTH
THTHH
HTHHT
THHTH
Eg how many ways can you get 2 heads out of 5 tosses?
THHHT
This is reflected by the fact that Pascal’s triangle is symmetrical
Why are these the same?
If r = n – r, the calculation gives the same result
ie substitute r = n – r in the rule for nCr
Number of combinations

5. Total number of possible arrangements
Where does the rule come from?
For example, why does ?
ie how many ways can you obtain 3 heads and 2 tails with 5 coins?
Call the 3 heads H1, H2 and H3 and the two tails T1 and T2
Any of the 5 could be the first you toss
Any of the remaining 4 could be the second
Etc until last position
Hence 5 x 4 x 3 x 2 x 1 = 5! ways to arrange the 5 coins
But we don’t care which Head came 1st, 2nd or 3rd and which tail came 1st or 2nd
There are 2! ways to order the tails and 3! ways to order the heads, giving 2! x 3! ‘duplicates’ of HHHTT
Consider the ways to obtain HHHTT:
The same is true for any other ways of obtaining 3 heads and 2 tails (eg HHTHT)
It follows that if = the number of distinct outcomes:
Duplicates
Desired, distinct outcomes
Total number of possible arrangements

6. Firstly: n different objects can be arranged in n! ways
Notes5 summary
Firstly: n different objects can be arranged in n! ways
Secondly: n objects with r of one type and (n - r) of another can be arranged in 𝒏𝑪𝒓= 𝒏 𝒓 = 𝒏! 𝒓! 𝒏−𝒓 ! ways

7. RRBB BBRR RBBR BRRB RBRB BRBR
WB 1
Find all the possible arrangements of
Three objects where one is red, one is blue and one is green
Four objects where two are red, and two are blue
First object can be chosen in 3 ways; 2nd object can be chosen in two ways and the third object can be chosen in one way
3!=3×2×1=6 𝑤𝑎𝑦𝑠
RBG RGB BRG BGR GRB GBR
b) If we treat the objects as four different things 𝑅 1 , 𝑅 2 , 𝐵 1 , 𝐵 2 then there are 4!=24 ways. However, arrangements with 𝑅 1 𝑅 2 are identical to those with 𝑅 2 𝑅 1 … and similarly with 𝐵 1 𝐵 2 and 𝐵 1 𝐵 2
So there are 4! 2!2! =6 𝑤𝑎𝑦𝑠
RRBB BBRR RBBR BRRB RBRB BRBR

8. A child has 4 yellow and 6 black identically shaped wooden bricks.
WB 2
A child has 4 yellow and 6 black identically shaped wooden bricks.
The bricks are assembled into a vertical tower one brick deep.
Find the number of different patterns of the bricks which can be formed
Ten bricks with 4Y and 6B can arranged in 𝟏𝟎! 𝟒!𝟔! = 210 ways

9. A bag has three £1 coins, two 50p coins and four 10p coins.
WB 3
A bag has three £1 coins, two 50p coins and four 10p coins.
Three coins are randomly selected. Find the probability that the value of the coins is a) £ b) £ c) 30p
£2.50 must be made of two x £1 and one 50p → 𝟑! 𝟐!𝟏! = 3 ways
The probability is given by:
P(£1) x P(£1/ £1 drawn) x P(50p/ two £1 coins drawn) x 3 ways
= 3 9 × 2 8 × 2 7 ×3= 1 14
b) £1.60 must come from £1, 50p and 10p → 3! ways
The probability is given by
3 9 × 2 8 × 4 7 ×3!= 2 7
c) 30p must come from 10p, 10p and 10p → 𝟑! 𝟑! = 1 way
The probability is given by
4 9 × 3 8 × 2 7 ×1= 1 21

10. Practice
Ex 1A

11. One thing to improve is –
KUS objectives BAT use factorial notation to find the number of arrangements of some objects
self-assess
One thing learned is –
One thing to improve is –

12. END