Repetition: Permutations Tutorial

A basic definition of permutation is: a set of distinct objects that is arranged in some sort of sequences, where order is important and the objects can only be used once. It is noted as n!, where n is the distinct objects, and n! means n(n – 1)(n – 2) … (n – n + 1). Also, an ordered arrangement of r elements of a set is called an r-permutation.

For example: A set, Q, has the members {1, 2, 3, 4, 5}. The arrangement of 5, 4, 3, 1, 2 is a permutation of set Q. The arrangement of 5, 4, 2 is called a 3-permutation of set Q, since there is only 3 elements, and so on.

The number of r-permutations of a set with n elements is denoted as P(n, r), where P(n, r) can be found by using the product rule. It is written as

P(n, r) = n(n – 1)(n – 2) ^… (n – r + 1) = n!/(n-r)!.

But with repetition, elements can be used repeatedly. For instance, to count how many ways can a string X – X – X can be formed from the 10 digits, 0 – 9, with repetition is allowed, you would have 10³ ways of forming the string. So, the number of r-permutations of a set of n objects with repetition allowed is

n^r.

So what does all this has to do with the real world, you asked? Well, for one thing, for the government in Nebraska to consider how many possible ways can they make the license plates and give it to the people would be a start. They would have to see how they put the letters and numbers on the plate to have enough for the whole Nebraska population. Consider the Nebraska license plate right now with the string X – X – X – N – N – N with repetition is allowed, where x is the letters of the alphabet and n is the numbers from 0-9. How many possible license plates could be produced? You would take 26³ times 10³, giving the result of 17,576,000 ways of making the license plates.

Enough said, how about you giving it a try?

How many strings of length 5 can be formed from the consonants in the English alphabet and the first letter of the string has to be Q, where repetition is allowed? (y is considered a consonant)

a) 4,084,101

b) 194,481

c) 2,441,880

d) 3,889,620

e) 1,094,560