Combinations Master CheatSheet - Easy Hacks

1. The "Match the Slots" Trick (The Golden Rule)
Forget the long formula with the confusing subtraction in the brackets.
Rule: The number of items on the top MUST match the number of items on the bottom.

If you need
C(10,3):

Top: Start at 10, count down 3 numbers (10×9×8).
Bottom: Start at 1, count up 3 numbers (1×2×3).
You don't need to write the rest of the factorials.

2. The "Mirror" Trick (The Lazy Rule)
This is the biggest time-saver in aptitude tests.
Rule: Choosing people to keep is the same as choosing people to throw away.

Formula:
C(n,k)=C(n,n−k)

Scenario: Find C(10,8).

The Hard Way: Writing out 8 numbers on top and 8 numbers on the bottom.
The Magic Way: If I pick 8 people to join the team from 10, that’s the exact same as picking 2 people to stay home.
Calculate
C(10,2) instead.
When to use: Whenever the "k" (the number you are choosing) is more than half of the total "n".
C(50,49)→C(50,1)=50
C(12,10)→C(12,2)
3. The "Handshake" Hack (Picking 2)
Aptitude tests love questions involving "handshakes" or forming "pairs" (which is just C(n,2)). Memorize this specific shortcut.

Rule:
n×(n−1)
​
If there are 20 people in a room and everyone shakes hands:

Take the number (20).
Multiply by the number below it (19).
Cut it in half.

4. The "Simplify Before You Multiply" Strategy
Never multiply the top numbers first! You will end up with huge numbers that are hard to divide. always Cancel out (reduce) first.

Example:
C(10,3)

Bad way:
10×9×8=720.
3×2×1=6.
720/6=120. (Too much mental energy).

Magic way (Cancelling):
See the 3 on the bottom? Cancel it with the 9 on top (leaving 3).
See the 2 on the bottom? Cancel it with the 8 on top (leaving 4).
Now you just have:
10×3×4=120.
5. The "Cheat Sheet" Patterns
Memorize these instant answers to save 30 seconds per question:

C(n, n) = 1
Question: Select 5 people from 5.
Answer: 1 way (Everyone comes).
C(n, 0) = 1
Question: Select 0 people from 5.
Answer: 1 way (Nobody comes).
C(n, 1) = n
Question: Select 1 person from 25.
Answer: 25 ways.