# Permutations & Combinations – Problems

## Permutations & Combinations – Problems

Example 1: A question paper is divided into a sections A, B, C containing 3, 4, 5 questions respectively. Find the number of ways of attempting 6 questions choosing at least one from each section.

Solution:

First Method: The selection of a question may be of the following

 Section (A) Section (B) Section (C) (3) (4) (5) 3 2 1 3 1 2 2 3 1 2 2 2 2 1 3 1 4 1 1 3 2 1 2 3 1 1 4

Total no. of ways of attempting 6 questions.

= ³C₃ ⁴C₂ ⁵C₁ + ³C₃ ⁴C₁ ⁵C₂ + ³C₂ ⁴C₃ ⁵C₁ + ³C₂ ⁴C₂ ⁵C₂ + ³C₂ ⁴C₁ ⁵C₃ + ³C₁ ⁴C₄ ⁵C₁ + ³C₁ ⁴C₃ ⁵C₂ + ³C₁ ⁴C₂ ⁵C₃ + ³C₁ ⁴C₁ ⁵C₄ = 805.

Second Method: Required No. of attempting 6 questions

= Total no. of arrangements – selection except question from c – selection except Q from A – selection except Q from B

= ¹²C₆ – ⁷C₆ – ⁹C₆ – ⁸C₆ = 805.

Example 2: Find the number of ways in which 4 letters can be put in 4 addressed envelopes so that no letter goes into the envelope meant for it.

Solution: Let E₁, E₂, E₃, E₄ are the envelopes corresponding to letters L₁, L₂, L₃, L₄. By the given condition the arrangement in the following.

There are 3 ways be keeping L₂ in E₁.

Similarly, there are 3 ways be keeping L₃ in E₁

Similarly, there are 3 ways be keeping L₄ in E₁

∴ The total number of ways = 3 + 3+ 3 = 9

 E₁ E₂ E₃ E₄ L₂ L₃ L₄ L₁ L₂ L₄ L₁ L₃ L₂ L₁ L₄ L₃