How many different arrangements can be made from all the letters of the word PAKISTAN?
- There are 8 letters, and A appears 2 times. All other letters appear once.
- Number of arrangements = 8!/2! = 40320/2.
- = 20160.
Answer: 20160 arrangements
Factorials and permutations
Maths · Permutation, combination and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
The fundamental principle of counting says that if one job can be done in m ways and a second job in n ways, the two together can be done in m × n ways. Every counting formula in this chapter comes from it.
A permutation is an arrangement of objects in a definite order, so AB and BA are different. The number of permutations of n different objects taken r at a time is written nPr. Factorial notation, n!, keeps the products short.
The count becomes smaller when some objects are alike, or when the objects are placed in a circle, because arrangements that look the same must be counted only once.
Rules to remember
- n! = n(n − 1)(n − 2) ... 3 × 2 × 1, and 0! = 1
- n! = n × (n − 1)!, so for example 9!/7! = 9 × 8
- nPr = n!/(n − r)! = n(n − 1)(n − 2) ... (n − r + 1), a product of r factors
- nPn = n! (all n different objects arranged in a row)
- Objects not all different: n!/(n1! × n2! × ...), where n1, n2, ... are the numbers of alike objects
- Circular permutations of n different objects: (n − 1)!
- Necklace or key ring (can be turned over): (n − 1)!/2
Common mistake
Students use n! for a round table. In a circle one object is fixed and the others are arranged round it, giving (n − 1)!, and a necklace that can be flipped gives half of that.
Worked example
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