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 |