Permutations and Combinations
Learn how to choose between P and C every single time. Master neighbor restrictions, "at least" selection cases, and circular arrangements without second-guessing.
The Core Concept
The Golden Rule of Counting: When selecting r items from a total of n items, your very first question must always be: Does the order of selection matter? If Order Matters ⟹ Permutation
P(n, r) (e.g., lining up, assigning specific titles, making codes). If Order Does NOT Matter ⟹ Combination C(n, r) (e.g., selecting a committee, drawing marbles, choosing a group).DSE Permutations and Combinations Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Permutations | P(n, r) = n! / (n − r)! | Order Matters: • Arrange • Line/Row • Code/Digits Used when rearranging the selected items results in a different, unique outcome. |
| 2. Combinations | C(n, r) = n! / [r!(n − r)!] | Order Does NOT Matter: • Choose • Committee • Group/Team Used when order is discarded. Note that: C(n, r) = P(n, r) / r! |
| 3. Neighbor Restrictions | • Must sit together • Cannot sit together | • Must sit together: Use the Tie Method. Group them as 1 block, arrange, then multiply by internal orders. • Cannot sit together: Use the Insert Method. Arrange others, then insert restricted items into the gaps. |
| 4. Selections with "At least" | • At least one… • At least two… | Use the Complementary Method:Total − Opposite cases |
| 5. Circular Arrangements | (n − 1)! | • Circular table • Key ring Fixing one person's position removes rotational symmetry (i.e., we divide n! by n). |
💡 Exam Traps & Secrets
⚠️ The Internal Arrangement OversightWhen using the Tie Method to keep items together, students almost always forget to multiply by the internal order of the tied block. Example: If 3 boys and 4 girls stand in a row, and the 3 boys must stand together:
1. Treat the 3 boys as 1 single block. You now have
2. You must multiply by the internal arrangements of the 3 boys inside their block ⟹
3. Correct answer:
1. Treat the 3 boys as 1 single block. You now have
1 block + 4 girls = 5 units to arrange ⟹ 5!.2. You must multiply by the internal arrangements of the 3 boys inside their block ⟹
3!.3. Correct answer:
5! × 3! (not just 5!).⚠️ The "At Least" Complementary TrapMany students assume that the complement of "at least" is always "none". This is only true for "at least one". Rule: If you are choosing a committee of 4 and need "at least 2 women," the opposite (complementary) cases are: 0 women AND 1 woman.
Tip: Sometimes, calculating the valid cases directly is much faster and safer than using the complement:
Tip: Sometimes, calculating the valid cases directly is much faster and safer than using the complement:
Cases = (2W and 2M) + (3W and 1M) + (4W and 0M)⚠️ Slot-Filling Method for Fixed PositionsIf certain positions are locked, fill those positions first before calculating the remaining arrangements. Example: If 2 specific boys must stand at the two ends of a row of 9 people, arrange those 2 boys first (
2!), then arrange the remaining 7 people in the middle (7!). Total ways = 2! × 7!.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Permutations with fixed position restrictions)5 boys and 4 girls are arranged in a row. How many ways can they be arranged if the 2 tallest boys must stand at the two ends of the row?
(Combinations with "at least" restriction cases)A committee of 4 is chosen from 7 men and 5 women. How many ways can this be done if the committee must contain at least 2 women?
(Circular permutation basics)In how many ways can 6 people be seated around a circular table?
Permutations and Combinations Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Permutation, Combination and Probability · Form 5
25 MC questions · 9 long questions
Frequently Asked Questions
Ask: does the order of selection matter? Arranging in a line, filling labelled positions, or forming sequences → use P(n,r). Choosing a group where order does not matter → use C(n,r). Remember: C(n,r) = P(n,r)/r! because combinations discard the r! internal orderings of the selected items.
Use the Tie Method: treat the restricted group as a single unit. Count arrangements of the reduced set (unit + remaining), then multiply by the internal arrangements of the tied group. For "cannot sit together," use the Insert Method: arrange the unrestricted items first, then insert the restricted items into the gaps between them.
Use the complement: Total unrestricted − None of the required group. This is faster than summing over "exactly 1", "exactly 2", … cases separately. However, for "at least k" where k > 1, the complement includes multiple cases — sometimes direct case calculation is safer.