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Counting the options
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An explanation of how to count the number of options in counting and probability.
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Counting the options:Consider the words A ,, CAT and HELP , if we want to know how many possible ‘words’ can
be created by rearranging the letters we could list all the possibilities:
For A :
A
There is only 1 option.
For ¿:
¿ TI
2 options 24 options are possible.
The longer the word the more tedious this
For CAT:
method becomes, instead we are able to
CAT ACT TAC come up with a more elegant solution by
CTA ATC TCA reflecting on how we come up with each of
6 options the ‘words’ on the lists.
Continuing to look at the word HELP :
For HELP:
HELP EHLP LEHP PELH For the choice of the first letter of the
HEPL EHPL LEPH PEHL word we have 4 options ( H , E , L or P ),
HLEP ELHP LHEP PLEH assuming we start with H , when we look
HLPE ELPH LHPE PLHE at the second letter we now only have 3
HPEL EPHL LPHE PHLE letters to choose from as H has already
HPLE EPLH LPEH PHEL been used ( E , L or P ), selecting E for our
second letter leaves us with 2 options for
third position and for the final place only 1 letter will remain.
So in total there are 𝟒 × 𝟑 × 𝟐 × 𝟏 = 24 potential words that can be formed.
This is called the Fundamental counting principle:
If there are 𝒂 ways that one event can be performed, 𝒃 ways that a second event can be performed,
𝒄 ways that a third event can be performed and so on, then there are 𝒂 × 𝒃 × 𝒄 × … ways in total
that the events can be performed successively.
Illustrative Example: repeated use of letters permitted
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