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04 Jun 2026, 00:32
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GMAT COUNTING MASTER DECISION TREE
Step 1: Are the objects DISTINCT or IDENTICAL?
DISTINCT = different fruits, different books, different flags, named people
IDENTICAL = identical candies, identical balls, identical cookies
Step 2: Am I DISTRIBUTING, ARRANGING, or SELECTING?
Distributed among → Distribution
Arrange, line up, seat → Permutation
Choose, select, committee → Combination
Step 3: Check restrictions
Can recipients receive multiple objects?
Must everyone receive at least one?
Does order matter?
MOST TESTED GMAT FORMULAS
Distribution of distinct objects:
k^n
Example:
5 fruits among 4 children
4^5
Distribution of distinct objects with at most one per recipient:
kPn
Example:
3 prizes among 5 students
5P3
Distribution of identical objects:
(n+k−1)C(k−1)
Example:
5 identical candies among 4 children
8C3
Distribution of identical objects with everyone receiving at least one:
(n−1)C(k−1)
Example:
5 identical candies among 4 children
4C3
Arrange all:
n!
Example:
5 students in a row
5!
Arrange some:
nPr
Example:
Arrange 4 of 6 flags
6P4
Choose only:
nCr
Example:
Choose 4 of 10 students
10C4
Repeated letters:
n!/(a!b!c!...)
Example:
PEPPER
6!/(3!2!)
Optional items:
2^n
Example:
7 condiments
2^7
HOW TO IDENTIFY n AND k
For k^n:
Ask:
How many choices does ONE object have?
= k
How many objects are making that choice?
= n
Example:
5 fruits among 4 children
One fruit has 4 choices.
There are 5 fruits.
Answer:
4^5
Example:
6 letters mailed to 3 mailboxes
One letter has 3 choices.
There are 6 letters.
Answer:
3^6
Example:
10 books placed on 4 shelves
One book has 4 choices.
There are 10 books.
Answer:
4^10
GMAT TRIGGER WORDS
Distributed among
→ Think k^n first
Check restrictions
Arrange, seat, line up, order
→ n! or nPr
Choose, select, committee, team
→ nCr
Optional, include or not include
→ 2^n
Identical, indistinguishable
→ Stars and Bars
Different, distinct, named
→ Distinct-object formulas
04 Jun 2026, 01:41
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some incremental understanding -
TYPE A: DISTINCT OBJECTS → DISTINCT RECIPIENTS
A1. Distribution with unlimited capacity
Formula:
k^n
n = number of objects
k = number of recipients
Meaning:
Each object independently chooses one recipient.
GMAT Example:
5 different fruits distributed among 4 children.
Answer:
4^5
A2. Distribution with at most one object per recipient
Formula:
kPn
Meaning:
After assigning an object to a recipient, that recipient becomes unavailable.
GMAT Example:
3 different prizes distributed among 5 students.
No student can receive more than one prize.
Answer:
5P3
= 5×4×3
A3. Distribution where everyone gets at least one
Formula:
Use Inclusion-Exclusion
General Formula:
k! × S(n,k)
(Stirling Number of the Second Kind)
GMAT usually expects Inclusion-Exclusion.
GMAT Example:
3 different fruits distributed among 2 children.
Each child gets at least one fruit.
Total:
2^3 = 8
Invalid:
All fruits to Child 1 = 1
All fruits to Child 2 = 1
Answer:
8−1−1
= 6
Note - this formula is applicable only when recipient are 2 people
k^n - n
A4. Distribution where each recipient gets exactly one object
Formula:
n!
GMAT Example:
5 different gifts assigned to 5 children.
Answer:
5!
TYPE B: IDENTICAL OBJECTS → DISTINCT RECIPIENTS
B1. Recipients may receive zero
Stars and Bars
Formula:
(n+k−1)C(k−1)
n = objects
k = recipients
GMAT Example:
5 identical candies among 4 children.
Answer:
(5+4−1)C(4−1)
= 8C3
B2. Everyone gets at least one
Stars and Bars
Formula:
(n−1)C(k−1)
Idea:
Give 1 object to each recipient first.
GMAT Example:
5 identical candies among 4 children.
Each child gets at least one.
Answer:
(5−1)C(4−1)
= 4C3
= 4
TYPE : YES/NO CHOICE PROBLEMS
E1. Each item may be included or excluded
Formula:
2^n
Meaning:
Each item has 2 choices:
YES or NO
GMAT Example:
7 condiments.
Any number may be chosen.
Answer:
2^7
TYPE A: DISTINCT OBJECTS → DISTINCT RECIPIENTS
A1. Distribution with unlimited capacity
Formula:
k^n
n = number of objects
k = number of recipients
Meaning:
Each object independently chooses one recipient.
GMAT Example:
5 different fruits distributed among 4 children.
Answer:
4^5
A2. Distribution with at most one object per recipient
Formula:
kPn
Meaning:
After assigning an object to a recipient, that recipient becomes unavailable.
GMAT Example:
3 different prizes distributed among 5 students.
No student can receive more than one prize.
Answer:
5P3
= 5×4×3
A3. Distribution where everyone gets at least one
Formula:
Use Inclusion-Exclusion
General Formula:
k! × S(n,k)
(Stirling Number of the Second Kind)
GMAT usually expects Inclusion-Exclusion.
GMAT Example:
3 different fruits distributed among 2 children.
Each child gets at least one fruit.
Total:
2^3 = 8
Invalid:
All fruits to Child 1 = 1
All fruits to Child 2 = 1
Answer:
8−1−1
= 6
Note - this formula is applicable only when recipient are 2 people
k^n - n
A4. Distribution where each recipient gets exactly one object
Formula:
n!
GMAT Example:
5 different gifts assigned to 5 children.
Answer:
5!
TYPE B: IDENTICAL OBJECTS → DISTINCT RECIPIENTS
B1. Recipients may receive zero
Stars and Bars
Formula:
(n+k−1)C(k−1)
n = objects
k = recipients
GMAT Example:
5 identical candies among 4 children.
Answer:
(5+4−1)C(4−1)
= 8C3
B2. Everyone gets at least one
Stars and Bars
Formula:
(n−1)C(k−1)
Idea:
Give 1 object to each recipient first.
GMAT Example:
5 identical candies among 4 children.
Each child gets at least one.
Answer:
(5−1)C(4−1)
= 4C3
= 4
TYPE : YES/NO CHOICE PROBLEMS
E1. Each item may be included or excluded
Formula:
2^n
Meaning:
Each item has 2 choices:
YES or NO
GMAT Example:
7 condiments.
Any number may be chosen.
Answer:
2^7
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