COUNTING RULES

Counting Rules make the calculation of odds or probabilities a lot easier.  They help us organize events by informing us of the number of possible ways in which we can arrange them.  In class, we will cover the following:

Fundamental Counting Rule: counts the number of sequences that are possible when matching the number of outcomes of one event with the outcomes of any number of other events.  For example, if you own 3 slacks, 4 shirts, 2 pairs of shoes, 2 pairs of socks, 1 undergarment and 2 coats, then you can combine your dresswear into 3x4x2x2x1x2 = 96 distinct outfits.

The Factorial Rule (!): counts the number of sequences that are possible when matching sequences of numbers in arrays of length equal to the number of possible outcomes.  For example, 5! = 5x4x3x2x1=120 = five numbers can be arranged in 120 different 5-tuples.

The Permutations Rule (nPr): counts the number of sequences that are possible when matching distinct sequences of numbers in arrays of shorter/lesser length than the number of possible outcomes, in which order matters.  For example, 5P2 = 5x4 = 20 = there are 20 ways of arranging numbers in pairs (sequences of length 2) when one can choose from five distinct numbers, in which order matters.

The Combinations Rule (nCr): counts the number of sequences that are possible when matching distinct sequences of numbers in arrays of shorter/lesser length than the number of possible outcomes, in which order does not matter.  For example, 5C2 = 5x42 = 10 = there are 10 ways of arranging numbers in pairs (sequences of length 2) when one can choose from five distinct numbers, in which order does not matter.

 More detail on Counting Rules

 I.                     Fundamental counting rule: the number of possible sequence-arrangements of joint compound events equals the product (multiplication) of the number of arrangements of each component/part.           

For example, if a car model can be offered to customers in 4 interior colors and 8 exterior colors, then the total number of car arrangements (by interior-exterior) is 4*8 = 32.

 

 

II.                   Factorial counting rule: the number of possible arrangements of distinct sequences of n objects into n-tuples is equal to n! (which reads "n factorial")

For example, how many quartets of the below objects can be formed so that each object only appears once in the 4-tuple.  Answer : 4! = 4*3*2*1 = 24 ways.

 

 

 

Or

 

 

Or

Or

The first object can be selected to fit in any of four 4 spots in the sequence; once a spot is selected for it, it cannot be used again in the sequence (because of the distinct nature of the sequences), and neither can the spot reserved for it be employed by another symbol.

 

 

 

 

or

or

The second object can be selected to fit in any of three 3 remaining spots in the sequence; once a spot is selected for it, it cannot be used again in the sequence (because of the distinct nature of the sequences), and neither can the spot reserved for it be employed by another symbol.

 

 

 

 

Or

 

 

The third object can be selected to fit in any of two 2 remaining spots in the sequence; once a spot is selected for it, it cannot be used again in the sequence (because of the distinct nature of the sequences), and neither can the spot reserved for it be employed by another symbol.

 

 

 

 

The fourth object can be selected to fit in the only (1) remaining spot in the sequence.

        Thus, applying the fundamental counting rule, we get that the total number of arrangements is 4x3x2x1 = 24.  This special pattern of counting is called factorial because we are "factoring out" one possibility after every consecutive selection in the sequence --until we exhaust the list of items being counted.  This accomplishes our objective to have every item of the sequence be different from every other one item in the list (distinct), while accounting for all items on the list.

 III.                  Permutations counting rules (shuffles): count the number of distinct sequences of items picked from a larger list of possibilities.  A permutations count is essentially a factorial count cut short because not all possible items get selected.

For example, "count the number of was in which 2 items can be picked out of a list of four items in such a way that the order of the sequence is important."  Answer: 4P2 = 4 x 3 = 12 ways. 

 

 

 


Or 

 

Or

 

Or

 

The first object can be selected from any of four 4 available; once a spot is occupied by one of the items, it cannot be used again; neither can the object repeat itself in the sequence (because of the distinct nature of the sequences).

 

 

 

 

or

or

And the second picked object can be any of three 3 remaining choices.

 

IV. Combinations counting rule:  count the number of distinct sequences of items picked from a larger list of possibilities in such a way that the order of appearance of the elements does not "matter" (affect the total count of sequences).

For example, "count the number of ways in which 2 items can be selected form a list of four items in such a way that the order of appearance of items in a sequence does not matter.  Answer: 4C2 = (4P2) 2! = (4 x 3) 2 = 6 ways.

 

 

 

                                                                        3 ways =

 

                                                           +          2 ways =

 

                                                            +          1 way =

 

                                                      adds up to 6 ways.