Counting: Permutations and Combinations Practice
The key question in counting is whether order matters and whether things are distinguishable. Be clear about what you are counting before applying a formula.
Basic, Standard: Grade 10 · Term 1 / Advanced: Grade 12 · Exam prep
Math problem generator
Permutations and combinations
Use permutations when order or roles matter (president and vice-president) and combinations when only the selection matters (choosing a committee). With repetition allowed, there are arrangements.
Circular arrangements, repeated items, shortest paths
Circular: Necklace: With repeated items:
Items that must be together are treated as one block; items that must be apart are placed into the gaps after arranging the others. Shortest grid paths are arrangements of right and up moves: across and up give paths.
Grouping and combinations with repetition
Putting 6 people two per room into rooms A, B and C gives ways, but splitting them into three unlabeled pairs gives . Divide by the arrangements of groups of equal size when the groups are unlabeled.
Nonnegative integer solutions of correspond to arrangements of circles and 2 bars, so there are of them.
Worked examples
A large die and a small die are thrown together. Count the outcomes in which the difference of the numbers is 3.
Hint
Make a table, or list the pairs that satisfy the condition.
Answer
Solution
Listing the pairs (large, small):
Counting,
In how many ways can 9 people be split into three groups of 3?
Hint
Count as if the groups were labeled, then divide by the number of ways to order groups of equal size.
Answer
Solution
If the groups were labeled,
The groups are unlabeled, so divide by :
How many triples of nonnegative integers satisfy ?
Hint
Think of arrangements of circles and bars (combinations with repetition).
Answer
Solution
Each solution corresponds to an arrangement of 10 circles and 2 bars:
Practice problems
Using three different digits from the 6 digits 0, 1, 2, 3, 4, 5, how many three-digit multiples of 5 can be formed?
Hint
Remember the hundreds digit cannot be 0. Fill the restricted positions first.
Answer
Solution
If the units digit is 0:
If the units digit is 5, the hundreds digit avoids 0 and 5, and the tens digit uses what is left:
In total,
In how many ways can 6 people be put into three rooms A, B and C, two per room?
Hint
Count as if the groups were labeled, then divide by the number of ways to order groups of equal size.
Answer
Solution
Choose 2 for A, then 2 for B; the rest go to C:
Using four different digits from 0, 1, 2, 3, 4, how many four-digit numbers that are multiples of 3 can be formed?
Hint
A number is a multiple of 3 exactly when its digit sum is. First choose the set of digits.
Answer
Solution
The sets of 4 digits whose sum is a multiple of 3 are
A set containing 0 gives numbers (0 cannot lead); otherwise . In total,