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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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Always ask these 2 questions first:
Q1. Are the objects distinct or identical?
CASE 1: Identical Objects + Distinct Boxes
Signal Words
Stars and Bars
Formula
If n identical objects are distributed among k distinct boxes:
Empty boxes allowed:
\binom{n+k-1}{k-1}
No empty boxes:
\binom{n-1}{k-1}
GMAT Example
5 identical apples distributed among 4 children.
Answer:
C(5+4−1, 4−1)
= C(8,3)
= 56
Mental Picture
You only care:
Child 1 gets how many?
Child 2 gets how many?
etc.
Not which apple goes where.
CASE 2: Distinct Objects + Distinct Boxes
Signal Words
Each object chooses a box
Formula
Empty boxes allowed:
k^n
GMAT Example
5 different fruits distributed among 4 children.
Each fruit has 4 choices.
Answer:
45 = 1024
Mental Picture
Apple chooses child.
Mango chooses child.
Banana chooses child.
Each object independently picks a destination.
CASE 3: Distinct Objects + Identical Boxes
Signal Words
Partitions (Stirling Numbers)
Formula
At most k identical groups:
S(n,1)+S(n,2)+...+S(n,k)
GMAT Example
5 different fruits distributed among 4 identical baskets.
Answer:
S(5,1)+S(5,2)+S(5,3)+S(5,4)
= 1+15+25+10
= 51
Mental Picture
You are creating groups.
{Apple, Mango}, {Banana}
is the same as
{Banana}, {Apple, Mango}
because baskets have no names.
Key Test
If swapping two boxes does NOT create a new arrangement → boxes are identical.
CASE 4: Identical Objects + Identical Boxes
Signal Words
Integer Partitions
GMAT Example
5 identical apples distributed among identical baskets.
Possible distributions:
5
4+1
3+2
3+1+1
2+2+1
2+1+1+1
1+1+1+1+1
Answer = 7
Mental Picture
Only basket sizes matter.
You are splitting a number.
Fast Identification Table
10-Second GMAT Decision Tree
Step 1:
Can I distinguish the objects?
YES → Distinct Objects
NO → Identical Objects
Step 2:
Can I distinguish the boxes/groups/children?
YES → Distinct Boxes
NO → Identical Boxes
Step 3:
Use:
Q1. Are the objects distinct or identical?
- Distinct = Apple, Mango, Banana (different)
- Identical = 5 same apples
- Distinct = Child 1, Child 2, Child 3
- Identical = Unlabeled baskets/boxes/groups
CASE 1: Identical Objects + Distinct Boxes
Signal Words
- Identical balls/apples/books
- Different children/boxes/bins
Stars and Bars
Formula
If n identical objects are distributed among k distinct boxes:
Empty boxes allowed:
\binom{n+k-1}{k-1}
No empty boxes:
\binom{n-1}{k-1}
GMAT Example
5 identical apples distributed among 4 children.
Answer:
C(5+4−1, 4−1)
= C(8,3)
= 56
Mental Picture
You only care:
Child 1 gets how many?
Child 2 gets how many?
etc.
Not which apple goes where.
CASE 2: Distinct Objects + Distinct Boxes
Signal Words
- Different fruits/books/students
- Different children/boxes
Each object chooses a box
Formula
Empty boxes allowed:
k^n
GMAT Example
5 different fruits distributed among 4 children.
Each fruit has 4 choices.
Answer:
45 = 1024
Mental Picture
Apple chooses child.
Mango chooses child.
Banana chooses child.
Each object independently picks a destination.
CASE 3: Distinct Objects + Identical Boxes
Signal Words
- Different fruits/books/students
- Identical baskets/groups
Partitions (Stirling Numbers)
Formula
At most k identical groups:
S(n,1)+S(n,2)+...+S(n,k)
GMAT Example
5 different fruits distributed among 4 identical baskets.
Answer:
S(5,1)+S(5,2)+S(5,3)+S(5,4)
= 1+15+25+10
= 51
Mental Picture
You are creating groups.
{Apple, Mango}, {Banana}
is the same as
{Banana}, {Apple, Mango}
because baskets have no names.
Key Test
If swapping two boxes does NOT create a new arrangement → boxes are identical.
CASE 4: Identical Objects + Identical Boxes
Signal Words
- Identical balls/apples
- Identical boxes/baskets
Integer Partitions
GMAT Example
5 identical apples distributed among identical baskets.
Possible distributions:
5
4+1
3+2
3+1+1
2+2+1
2+1+1+1
1+1+1+1+1
Answer = 7
Mental Picture
Only basket sizes matter.
You are splitting a number.
Fast Identification Table
| Objects | Boxes | Method |
|---|---|---|
| Identical | Distinct | Stars & Bars |
| Distinct | Distinct | kn (assignment) |
| Distinct | Identical | Stirling / Partitions |
| Identical | Identical | Integer Partitions |
10-Second GMAT Decision Tree
Step 1:
Can I distinguish the objects?
YES → Distinct Objects
NO → Identical Objects
Step 2:
Can I distinguish the boxes/groups/children?
YES → Distinct Boxes
NO → Identical Boxes
Step 3:
Use:
- Identical + Distinct → Stars & Bars
- Distinct + Distinct → kn
- Distinct + Identical → Partitions/Stirling
- Identical + Identical → Integer Partitions
Moderator:
