I understand you are taking a college course in mathematics and studying permutations and combinations. Permutations and Combinations date back through the ages. According to Thomas & Pirnot (2014), there is evidence of these mathematical concepts as early as AD 200. As we solve some problems you will see why understanding these concepts is important especially when dealing with large values.
I also understand you are having problems understanding their subtle differences, corresponding formulas nPr and nCr and the fundamental counting principle. Before we review some exercises, I would like to provide you with some definitions you will need in solving some problems.
According to Thomas & Pirnot (2014), a permutation is an ordering of distinct objects in a straight line. If we select r different objects from a set of n objects and arrange them in a straight line, this is called a permutation of n objects taken r at a time. The number of permutations of n objects taken r at a time is denoted by P (n,r).
In working with permutations and combinations we are choosing r different objects from a set of n objects. The big difference is whether the order of the objects is important. If the order of the objects matter, we are dealing with permutations. If the order does not matter, then we are working with combinations (Thomas & Pirnot, (2014), p. 624, 626).
According to Thomas & Pirnot (2014), in combinations if we are choosing r objects from a set of n objects and are not interested in the order of the objects, then to count the number of choices, we must divide P (n,r)* by r!. We now state this formally: Formula for computing C(n,r). If we chose r objects from a set of n objects we say we are forming a combination of n objects ta...
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...someone who is not the owner of the IPad will guess the code?
Solution:
36C4=36x35x34x33=1,413,720
There are 10 numeric digits on a keypad and 26 letters. Adding these we get 36 digits representing the first code, 35 digits are used for the second code, 34 digits are used for the third code and 33 digits for the fourth code. Leaving us with 1,413,720 possible pass code combinations someone other than the owner can use. The likelihood of all these combinations being used is slim.
These counting principles are important in understanding odds. For example, I rarely play the lottery but have relatives who play on a regular basis. Mathematically the odds are slim, however the game can only be won by chance. I hope the definitions and the equations have brought clarity to the subject.
Reference
T. L. Pirnot.. (2014). Mathematics all around. (5th ed.). Boston, MA.
In other words T(n) can be expressed as sum of T(n-1) and two operations using the following recurrence relation:
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By using combinations from the above – But this approach has to be made with caution, because confusion can appear in the message.
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and 8 can be written as 2 , while 5, 6, and 7 can be written using some