When do we add and when do we multiply?

1
When do we add and when do we multiply?
2
How many ways can we draw a king or a seven from
a deck?

• There are 4 kings and 4 sevens. Each of these
  cards satisfies the event.
• So, 4 4 8

How many ways can we draw a king and a seven from
a deck?
There are 4 kings to draw first, then for each
king, there are 4 sevens that can match with it.
Each event has a pair. So, 4 X 4 16
3
Combinations
4
Election 1

• How many ways can a President, Vice President,
  and Secretary be selected from 4 people?

4
3
2
n 4 X 3 X 2 24
This is a Permutation
5
Election 2

• How many ways can a three person general council
  be created from four people?

4
3
2
One selection process (using permutations) could
have resulted in John Mary Susan
6
Another selection could be Mary Susan John
Or Susan Mary John
If all that is being generated are groups of
three, then there is no difference between
these (Like drawing a 5 card poker hand..)
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The permutation counting technique counts every
order. Order does not matter when a general group
(or Subset) is being selected. Therefore, we can
not use permutations when the order of the
selected items does not matter.
Our count will be too high, so we will use
division to reduce our count.
8
How many ways can Mary John and Susan be ordered?
3
2
1

• n 3 X 2 X 1 6

We have counted the same case 6 times
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To offset this, divide the number of permutations
by 6 (6 / 6 1 case counted).(this allows us
to keep our permutation/ factorial structure)
6 (notice), happens to be 3!, where 3 is the
length of the subset.
10

• We are going to use our permutation formula with
  a slight adjustment in the denominator.
• This adjustment will reduce the total count by
  the correct amount.

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Combination

• A combination is a collection of chosen objects
  for which order does not matter.
• C(n,r), nCr, or n , represent the
• number of combinations possible in which r
  objects are selected from a set of n different
  objects.

r
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Think of poker

• 52P5
• 52C5

Recognize the difference
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• C(n,r) n!
• (n r)!r!

The only difference between the permutation
formula and the combination formula is the extra
divisor of r!
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How many different sampler dishes with 3
different flavours could you get at an ice cream
shop with 31 different flavours?
n 31 and r 3
31!
C(31,3)
(31 3)!3!
4495
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