Permutations vs Combinations: The Quantitative Aptitude Guide
In Quantitative Aptitude, choosing between permutation and combination is the first step to solving counting problems. Let's compare them side-by-side.
head-to-Head Comparison
| Basis | Permutations | Combinations |
|---|---|---|
| Core Meaning | Arrangement of items where the order of arrangement matters (e.g., seating arrangements, lock codes) | Selection of items where the order of selection does not matter (e.g., forming committees, choosing cards) |
| Formula | $$^n P_r = \frac{n!}{(n-r)!}$$ | $$^n C_r = \frac{n!}{r!(n-r)!}$$ |
| Result Count | Always larger (or equal) to combinations for the same $n$ and $r$ | Always smaller (or equal) because order variants are consolidated |
| Key Words in Questions | "Arrange", "Line up", "Number plates", "Codes" | "Select", "Choose", "Committee", "Group" |
The 'Committee' Trap
If a question asks you to select 3 people from 10 to form a committee, use **Combinations**. If it asks you to select a President, VP, and Secretary (ordered roles), you must use **Permutations** because the roles make the order matter!
Common Ground (Similarities)
- Both are core counting principles in probability and statistics.
- Both utilize factorials and are bound by the condition $0 \le r \le n$.
Test Your Understanding
Q1: How many ways can you arrange 3 books from a shelf of 5?
10
20
60 ✅
120
Explanation: Since it is an 'arrangement', order matters: $^5 P_3 = 5 \times 4 \times 3 = 60$.