The counting principle involves three concepts: the multiplication rule, combinations and permutations. The principle of multiplication states that, if there are n 1 outcomes of experiment
, n2 outcomes of experiment
and nm outcomes of experiment
then there are
outcomes of the whole composite experiment (Penn State Eberly College of Science, 2018). Permutation refers to all the possible arrangements of objects or numbers where order matters and if order is compromised, a different permutation is obtained. An example of permutation is the selection of telephone numbers. Combination on the other hand involves arrangement of objects or people where order does not matter and repetition is not essential as it results to a non-sensible outcome ( Tobias, 2011) . An example of a combination problem is when several people are to be chosen from a large group to form a small committee. Repetition in this case would mean one person is selected twice which obviously does not make sense. The three counting techniques have been used in real life situations and will be applied in solving the questions presented below in a Q & A format.
Question 1
Bella works at sub shop and wants to know how many different types of subs she can make considering all of the available options. There are six types of rolls, five types of meats, eight types of vegetables, and three types of cheeses. If two choices are made from each of the four categories, how many types of subs can Bella make?
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Solution
There are two choices for each category and in this case order doesn’t matter. Combinations could therefore be used in this problem. In essence there are
possible ways of choosing a roll,
possible ways of choosing meat,
ways of choosing vegetables and
ways of choosing chesses. If Bella has to make unique subs, then repetition is not allowed. Therefore there the number of subs Bella can make equals
.
Question 2
How many odd, five-digit numbers can be created from the digits 1 to 5 if repetition is allowed?
Solution
In this case, the last number could be 5,3, or 1 if the 5 digit number is odd. There are 3C1=3 ways of choosing the last digit. For the first letter any of the five could be used and there are 5C1=5 ways and similarly there are five ways of choosing the second, third, and fourth letters. Since order does not matter then the total number of ways an odd five digit number could be written are
.
Question 3
How many ways can four calculators are chosen for testing from a group of 10?
Solution
In this case, the order in which the calculators are selected is not important but one calculator cannot be chosen twice and therefore repetition is not allowed. The combination concept is therefore used.
, there are 210 ways four calculators can be chosen from a group of 10.
The three counting techniques: multiplication principle, combinations, and permutation are applicable in real life situations as those above. The multiplication rule multiplies the number of outcomes of each independent experiment. Permutation refers to the arrangement of events, people or numbers where order matters. Combination on the other hand is the number of unique arrangements where order is not important. Permutations and combinations appear similar but the distinguishing characteristic is that permutation allows repetition but combinations do not. The techniques are used to determine the number of possible arrangement outcomes of an event.
References
Penn State Eberly College of Science (2018). STAT 414/415: Probability Theory and Mathematical Principles; Counting techniques. Retrieved from https://newonlinecourses.science.psu.edu/stat414/node/28/ Accessed 10 August, 2019.
Tobias, P. A., & Trindade, D. (2011). Applied reliability . Chapman and Hall/CRC.
