🔢 Combination

Last Updated: Jan 2026

Combination deals with selection where order does NOT matter. In exams and real problems, combination questions appear in many forms, often hidden inside words like choose, select, form groups.

🗣 Hinglish Tip: Combination = sirf choose karna, arrangement se farq nahi padta

Formula

Selection of r objects from n distinct objects:

${}^nC_r = \frac{n!}{r!(n-r)!}$

Simple Combination (Basic Selection)

• Just selecting objects
• No restriction

Example

From 5 students, how many ways to choose 2 students?

${}^5C_2 = \frac{5!}{2! \times 3!} = 10$

2 Combination with Repetition Allowed

• Same object can be chosen more than once
• Common in distribution problems

Formula

${}^{n+r-1}C_r$

Example

How many ways to select 3 balls s of balls?

$(5 + 3 - 1)C_3 = {}^7C_3 = 35$

🗣 Hinglish Tip: Repetition allowed + selection = stars and bars

Combination with Restrictions

Case A: Specific Objects Must Be Selected

Example: From 6 students, how many groups of 3 contain A?

Logic:

• Fix A
• Choose remaining from 5

${}^5C_2$

Case B: Specific Objects Must NOT Be Selected

Example: From 6 students, how many groups of 3 do NOT contain A?

${}^6C_3 - {}^5C_3$

At Least / Questions

General Rule

At least = Total - Not allowed

Example

From 7 men and 3 women, how many committees of 4 contain at least 1 woman?

• Total committees:

${}^{10}C_4$

• Committees with no women:

${}^7C_4$

• Required:

${}^{10}C_4 - {}^7C_4$

Group Formation Problems

• Forming teams, committees, panels

Example

How many ways to form a team of 3 from 4 boys and 3 girls?

${}^7C_3 = \frac{7!}{3!4!} = 35$

Example with Condition

Team of 3 with 2 boys and 1 girl:

${}^4C_2 \times {}^3C_1$

Distribution Problems (Combination Based)

• Distributing identical items into distinct boxes
  Formula:

${}^{n+r-1}C_{r-1}$

Example:

• Distribute 5 identical balls into 3 boxes:

${}^7C_2 = 21$

Selection from Repeated Groups

Example

From 3 red, 4 blue, and 5 green balls, how many ways to choose 2 balls of different colors?

Logic Table

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