Master Combinations for CBSE Class 11 Applied Mathematics. Learn formulas, properties, and step-by-step solutions for counting and selection problems.
A combination is a selection of items from a collection such that the order of selection does not matter. Unlike permutations, where arrangement is key, combinations focus solely on the group composition. For example, selecting a committee of 3 members from a group of 10 is a combination problem.
Step 1: Identify n=5 and r=2.
Step 2: Apply formula ⁵C₂ = 5! / (2! · 3!).
Step 3: Calculate (5 · 4) / (2 · 1) = 10.
Answer: 10 ways.
Combinations follow specific algebraic properties that simplify complex calculations. The most important property is the symmetry rule, which states that choosing r items is equivalent to leaving behind n-r items. Another useful identity is Pascal's identity, which relates combinations of smaller sets to larger ones.
Step 1: Calculate ¹⁰C₈.
Step 2: Use property ¹⁰C₈ = ¹⁰C₁₀₋₈ = ¹⁰C₂.
Step 3: Calculate (10 · 9) / 2 = 45.
Answer: 45.
Often, problems impose conditions such as 'including a specific person' or 'excluding a specific object'. When an item must be included, we reduce both n and r by 1. When an item must be excluded, we reduce only n by 1.
Step 1: Select 3 members from 6, including 1 specific person.
Step 2: We need 2 more from 5 remaining: ⁵C₂.
Step 3: Calculate (5 · 4) / 2 = 10.
Answer: 10 ways.