Home/quantitative-aptitude/quantitative-aptitude

Permutation & Combination

Master the art of counting without counting. Learn the Slot Method, Grouping Tricks, and Rank Shortcuts to solve complex arrangement and selection problems with zero confusion.

1. The Fundamental Counting Principle

Before formulas, understand logic.

The AND Rule (Multiplication): If you must do Task A AND Task B, multiply their ways.
Example: 3 Shirts AND 2 Pants = $3 \times 2 = 6$ outfits.

The OR Rule (Addition): If you must do Task A OR Task B, add their ways.
Example: Go by Bus (3 types) OR Train (2 types) = $3 + 2 = 5$ ways.

Example:

Q: A room has 2 doors and 3 windows. In how many ways can a thief enter through a door and exit through a window?

Solution: Enter (2 ways) AND Exit (3 ways) =

2 \times 3 = 6

ways.

2. Permutation vs. Combination: The 'Order Test'

The biggest confusion is 'Which formula to use?'. Apply the Order Test:

1. Change the order of selected items.
2. Does it make a new outcome?
- YES $\to$ Order Matters $\to$ Permutation (Arrangement)
- NO $\to$ Order Doesn't Matter $\to$ Combination (Selection)

Keywords:
- Permutation: Arrange, Line up, Code, Rank, Position.
- Combination: Select, Choose, Team, Group, Handshake.

Example:

Q: Selecting 3 students for a photo vs. Selecting 3 students for a trip.

Solution: Photo: Order matters (Left/Right position changes photo)

\to

Permutation.
Trip: Order doesn't matter (Ram, Shyam, Geeta is same group as Geeta, Ram, Shyam)

\to

Combination.

3. The 'Slot Method' (Sniper Method)

Forget $nPr$ ! Use the Slot Method for arrangements.

Step 1: Draw empty slots (boxes) equal to the number of positions.
Step 2: Fill each slot with the number of options available for that position.
Step 3: Multiply them.

Visual Model:
Arranging 3 people on 5 chairs?
[5 options] $\times$ [4 options] $\times$ [3 options] = 60 ways.
(First person has 5 choices, next has 4, next has 3).

Example:

Q: How many 3-digit numbers can be formed using 1, 2, 3, 4, 5 without repetition?

Solution: 3 Slots: [ _ _ _ ]
1st Slot: 5 choices. 2nd Slot: 4 choices. 3rd Slot: 3 choices.
Total =

5 \times 4 \times 3 = 60

4. Grouping Method (String Method)

Use this when items must 'always keep together'.

The Trick:
1. Tie the 'together' items with a string and treat them as ONE unit.
2. Arrange the main units.
3. Multiply by the internal arrangement of the tied group.

Example:

Q: Arrange 'APPLE' so P's are together.

Solution: Units: [A], [L], [E], [PP]. Total 4 units

\to

4!

Inside [PP]:

\frac{2!}{2!} = 1

way.
Total =

4! \times 1 = 24

5. Gap Method (Items Never Together)

Use this when items must 'never sit together'.

The Trick:
1. Arrange the 'others' first (creates gaps).
2. Place the specific items in the Gaps (spaces between adjacent items + ends).

Visual:
_ O _ O _ O _ (3 Others create 4 Gaps).
Select gaps using $nCr$ , then arrange.

6. Circular Permutation Hack

In a circle, there is no 'start' or 'end'.

Rule 1 (People/Distinct Items): $(n-1)!$
- Fix one person to create a reference point, then arrange remaining $(n-1)$ .

Rule 2 (Necklace/Garland/Identical Flip): $\frac{(n-1)!}{2}$
- Clockwise and Anti-clockwise look same (flippable).

7. Dictionary Rank Shortcut

Find rank of word 'MAKE' in dictionary.
1. Write letters alphabetically: A, E, K, M.
2. Fix 'A' first $\to$ remaining 3! words. (Does target match? No, target starts with M).
3. Fix 'E' $\to$ 3! words.
4. Fix 'K' $\to$ 3! words.
5. Fix 'M' $\to$ Match! Now fix next letter 'A'...