Counting Strategies

Question 1

What are counting strategies?

Answer 1

Counting Strategies are different strategies to solve equations.

Question 2

Combinations & Permutations

Answer 2

When the order matters, it is a Combination, however when the order does matter, it is a Permutation. In other words, a permutation is an ordered combination.

There are two types of permutation:

• Repetition is allowed: such as a lock it could be ‘333’.
• No Repetition: for example the first three people in a running race. One person can not be first and second.

Permutations with repetition:
- These are the easiest to calculate.
- When a thing has n different types… …we have n choices each time.
- For example: Choosing 3 of those things, the permutations are:
n × n × n
( n multiplied 3 times)
- More generally: Choosing r of something that has n different types, the permutations are:
n × n × … ( r times )
- (In other words, there are n possibilities for the first choice, THEN there are n possibilities for the second choice, and so on, multiplying each time.)
- Which easiest to write down using an exponent of r:
n × n × … ( r times ) = n ( to the power of r )

The formula n ( to the power of r ):
- Where n is the number of things to choose from, and we choose r of them, repetition is allowed, and order matters.

Permutations without Repetition: Calculator Button: ( nPr )
- In this case, we have to reduce the number of available choices each time.
- Without repetition, our choices get reduced each time.
- We use “factorial function” or ‘!’ to write this mathematically.
- The factorial function (symbol: !) just means to multiply a series of descending natural numbers.
- 1! = 1 or 1 × 1
- 2! = 2 or 1 × 2
- 3! = 6 or 1 × 2 × 3
- 4! = 24 or 1 × 2 × 3 × 4
- 5! = 120 or 1 × 2 × 3 × 4 × 5
- 6! = 720 or 1 × 2 × 3 × 4 × 5 × 6
n!
——————————-
( n - r ) !
- Where n is the number of things to choose from, and we choose r of them, no repetition, order matters.

Notation:
- Instead of writing the whole formula, people use different notations.

Combinations: Calculator Button: ( nCr )

• There are also two types of combinations ( remember the order does not matter)
  - Repetition is allowed: Such as coins in your pocket
  - No Repetition: Such as lottery numbers

Combinations without Repetition:

• This is how lotteries work.
• The numbers are drawn one at a time, and if we have the lucky numbers ( no matter what order ) we win!
• The easiest way to explain it is to:
  - assume that the order does matter ( Permutations )
  - The alter it so the order does not matter.
• Permutations have more possibilities than Combinations.
• We just adjust our permutations formula to reduce it by how many ways the object could be in order ( because we aren’t interested in their order anymore ):

             n!                            1                         n!
  ---------------------      ×      --------    =     --------------------
        ( n - r ) !                        r                   r! ( n - r ) !

• That formula is so important it is often just written in big parentheses:

               n!                                       n    ---------------------------          =     (                )
        r! ( n - r ) !                                  r 

• Where n is the number of things to choose from, and we choose r of them, no repetition, the order doesn’t matter.
• It is often called “n choose r” ( such as “16 choose 3” ) And is also known as the ‘binomial coefficient-

Pascal’s Triangle:
- We can also use Pascal’s Triangle to find the values. Go down to row “n” ( the top row is 0 ), and then along “r” places and the value there is our answer.

Combinations with Repetition:
- Let’s say there are five flavours of ice cream: Banana, Chocolate, Lemon, Strawberry and Vanilla.
- We can have three scoops.
- How many variations will there be?
- Let’s use letters for the flavours ( b, c, l, s, v ).
- ( just to be clear there are n=5 things to choose from, we choose r=3 of them, the order does not matter, and we can repeat ).
- We can write this as…
( r + n - 1 ) ! r + n - 1
——————————- = ( )
r ! ( n - 1 ) ! r
- Where n is the number of things to choose from, and we choose r of them, repetition is allowed.

Summary:

Permutations (order matters):
- Repeats allowed: n (to the power of r)
- No Repeats: n!
—————-
( n - r ) !
Combinations (order doesn’t matter):
- Repeats allowed: ( r + n - 1 ) !
————————–
r! ( n - 1 ) !
- No Repeats: n!
——————————
r! ( n - r ) !