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Basic Combination Selection

Combination deals with selecting objects where order does NOT matter. The number of ways to choose 'r' objects from 'n' distinct objects is denoted as ⁿCᵣ or C(n,r) = n! / [r! × (n-r)!]. Combinations are used when forming committees, selecting teams, or any situation where the arrangement of selected items is irrelevant.

10Worksheets
200+Practice Questions
BeginnerDifficulty
2-3 hoursHours to Master

What You'll Learn

Introduction & Overview Why This Matters How to Solve Pro Tips & Tricks
Shortcut Methods Common Mistakes Practice Worksheets Exam Importance

Introduction to Basic Combination Selection

Combination deals with selecting objects where order does NOT matter. The number of ways to choose 'r' objects from 'n' distinct objects is denoted as ⁿCᵣ or C(n,r) = n! / [r! × (n-r)!]. Combinations are used when forming committees, selecting teams, or any situation where the arrangement of selected items is irrelevant.

Prerequisites

Factorial concept Permutation basics Division of factorials Fundamental Counting Principle
Why This Matters: Combination problems are extremely common in competitive exams. You can expect 2-3 questions in SSC CGL, 2-3 in Banking PO, and 2-3 in Railways RRB exams.

How to Solve Basic Combination Selection Problems

1

Step 1: Identify whether order matters (if not, use combination)

2

Step 2: Identify total objects (n) and number to select (r)

3

Step 3: Apply the formula: ⁿCᵣ = n! / [r! × (n-r)!]

4

Step 4: Simplify using the property: ⁿCᵣ = ⁿC_{n-r}

5

Step 5: Calculate by canceling common factors for easier computation

6

Step 6: For selection with restrictions, break into cases

7

Step 7: Verify the answer is an integer and less than the corresponding permutation

Pro Strategy: Always check if order matters. In combination problems, the selected items are treated as a set, not a sequence. Use the symmetry property ⁿCᵣ = ⁿC_{n-r} to simplify calculations (choose the smaller of r and n-r).

Example Problem

Example: From a class of 10 students, how many ways to select a committee of 3 students? Solution: Step 1: Order doesn't matter (committee members have no ranks) Step 2: n = 10, r = 3 Step 3: ¹⁰C₃ = 10! / (3! × 7!) Step 4: = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 Answer: 120 ways

Pro Tips & Tricks

  • ⁿCᵣ = ⁿC_{n-r} (useful for simplifying when r > n/2)
  • ⁿC₀ = ⁿCₙ = 1
  • ⁿC₁ = n
  • For calculation: ⁿCᵣ = [n × (n-1) × ... × (n-r+1)] / r!
  • ⁿC₂ = n(n-1)/2 (handshake formula)
  • ⁿC₃ = n(n-1)(n-2)/6

Shortcut Methods to Solve Faster

Use Pascal's Triangle for small values of n
Cancel factorials before multiplying to avoid large numbers
For selecting committees with minimum/maximum conditions, use sum of combinations

Common Mistakes to Avoid

Using permutation when combination is needed
Forgetting to divide by r! to account for order not mattering
Confusing ⁿCᵣ with ⁿPᵣ
Not using the symmetry property to simplify calculations

Exam Importance

Basic Combination Selection is an important topic for various competitive exams. Here's how frequently it appears:

SSC CGL
2-3 questions
BANKING PO
2-3 questions
RAILWAYS RRB
2-3 questions
CAT
1-2 questions
INSURANCE
2-3 questions

Ready to Master Basic Combination Selection?

Start with Worksheet 1 and work your way up to expert level! Each worksheet includes:

20 practice questions
Detailed solutions
Step-by-step explanations
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