Practice Combinatorial counting
Combinatorics counts possible arrangements or selections without listing each one.
The counting method depends on whether order matters and repetition is allowed.
Exercises with solutions
When counting possibilities two cases matter. With replacement every stage has the same number of options and you multiply the counts. Without replacement each stage has one option fewer. For all orders without replacement, n! = n · (n−1) · … · 1.
Exercise 1
How many possibilities does a three-digit lock with digits 0 to 9 have?
Show solution and working
1,000
Three positions with 10 options each, replacement allowed: 10 · 10 · 10 = 1,000.
Exercise 2
In how many ways can 3 children stand next to each other?
Show solution and working
6
Without replacement and with order: 3! = 3 · 2 · 1 = 6.
Exercise 3
2 children are chosen from 5 and the order does not matter. How many possibilities are there?
Show solution and working
10
First 5 · 4 = 20 ordered pairs. Since order does not matter, each selection was counted twice: 20 ÷ 2 = 10.
Exercise 4
In how many ways can 4 different books be arranged in a row?
Show solution and working
24
4! = 4 · 3 · 2 · 1 = 24.
Common mistake
Using the same number of options at every stage when there is no replacement. If 3 children are chosen one after another, the second child has only 2 places left, not 3.
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