Pascal

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Pascals Triangle
How do we expand (x1)5 quickly?
We can use Pascals triangle.
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The Factorial Function
n!1 x 2 x 3 xx (n-1) x n
Six cyclists enter a race. All finish at
different times. There is a first and second
prize.

• (a) How many different ways can the cyclists
  finish the race?
• How many different ways can prizes be awarded?

(a) There are 6 different possible winners.
For each winner there are 5 possible 2nd places.
For each of these there are 4 possible 3rd places.
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(b) From (a) there are 30 ways of getting 1st
and 2nd prizes.
This can also be answered using factorials.
This is a method of calculating the number of
ways of arranging r objects from a pool of n
objects. (2 from 6)
P refers to permutation.
(b) Required arranging 2 from 6.
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Ex.2 From a palette of 7 colours, you can pick
4. How many different ways can this be done?
The order of picking 4 does not matter here.
Red, blue green and yellow is the same as green
yellow red and blue.
In this example we are selecting 4 from 7 but the
order (arrangement) does not matter.
There are 4! Ways of arranging four colours.
This is a common problem (National Lottery)
C refers to combination.
This is selection without arrangement.
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Hence 4 from 7
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Hobsons Choice!
Hobson hired out horses. You paid your money and
got the horse of your choice. (As long as it was
the horse he offered you)
How many ways can you pick one horse when there
is only one horse to pick?
Now we know the answer is 1 so,
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What is the chance of winning the lottery?
Just a little further notation
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A common Fact.
Proof
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Ill let you investigate this !!
Page 7 Exercise 2B Questions 1(a), 2, 4(a)(b),
5(a), 6(a), 7(a)(b)(d). TJ Exercise 1
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The Binomial Theorem
When generalised,
This expansion is known as the Binomial Theorem
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We can use the binomial theorem to expand
brackets or to find a specific term in a
polynomial.
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MIA Page 9 Exercise 3A TJ Exercise 2 Question 1
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Further Expansions
Consider each set of brackets separately.
(trust me!!)
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It is best to do this in tabular form to avoid
errors
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Further practice on page 11 Ex 3B
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Approximation
Fact
We can use this fact to make useful
approximations.
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Page 13 Exercise 4 Page 14 Review