Permutation and Combination Practice Questions | Set 1

Permutations and combinations (PnC) are fundamental concepts in combinatorics used to count arrangements and selections of objects.

• Permutations refer to the different ways of arranging a set of items where order matters. For example, the arrangement of letters in a word or the order of runners finishing a race.
• Combinations, on the other hand, involve selecting items from a group without regard to the order of selection. This is used when the arrangement of items does not matter, such as choosing team members from a group.

Permutation and Combination Practice Questions

1. n students appear in an examination, find the number of ways the result of the examination can be announced.

Answer is 2ⁿ Examples: Input : n = 6 Output : Each student can either pass or fail in the examination. so ,there exists 2 possibilities for each of the 6 students in the result. hence total number of ways for the result=(2)⁶ Input : n = 8 Output :(2)⁸=256

2. 'n' matches are to be played in class a chess tournament, find the number of ways in which their results can be decided

Answer is (3)ⁿ ways Examples: Input : n = 3 Output: The results of each of the 3 matches can be three ways namely win ,draw or loss since total no. of ways in which results of 3 matches can be decided =(3)³ Input: 6 Output:(3)⁴=81

3. A badminton tournament consists of 'n' matches. (i) Find the number of ways in which their results can be forecasted are given. (ii) Total number of forecasts containing all correct results. (ii) Total number of forecasts containing all wrong results.

Answer (i) (2ⁿ) (ii) 1 (iii) 1 Examples: Input : A badminton tournament consists of 3 matches. (i) In how many ways can their results be forecast ? (ii) How many different forecasts can contain all correct results ? (iii) How many different forecasts can contain all correct results ? Output:(i) Each badminton match can be decided in only 2 ways either win or loss for a particular team so total number of ways results of 3 matches can be forecast=2³=8 (ii) Results of each match can be forecast wrong in only 1 way Total no. of forecasts containing all wrong results = (1³) = 1 (iii) Similarly, result of each can be forecast correct in only 1 way. total no .of forecasts containing all correct results = (1³) = 1

4. Find the number of ways in which 'n' different beads can be arranged to form a necklace

Answer is (n-1)!/2 Examples: For example 4 beads can be arranged in following ways. ....b1 b2.......b4 ....b3 ....b1 b3.......b2 ....b4 ....b1 b4.......b3 ....b2 Since it does not matter where we place first bead. Total ways to arrange is (n - 1)!. But clockwise and anticlockwise arrangements are same, so total arrangements are (n - 1)!/2

5. There are 'n' questions papers, find the no, of ways in which a student can attempt one or more questions

Answer: (2ⁿ-1)ways. For example a student will solve one or more questions out of 4 questions in following ways. 1) The student chooses to solve only one question, can choose in ⁴C₁ 2) The student chooses to solve only two questions, can choose in ⁴C₂ 3) The student chooses to solve only three questions, can choose in ⁴C₃ 3) The student chooses to solve all four questions, can choose in ⁴C₄ So total ways are ⁴C₁ + ⁴C₂ + ⁴C₃ +⁴C₄ =2⁴-1 = 15 ways We know sum of binomial coefficients from ⁿC₀ to ⁿC_n is 2ⁿ

More Practice Questions on Permutation and Combination

Unsolved Practice Problems on Permutation and Combination

• Problem: In how many ways can you arrange 3 books out of a set of 5 different books on a shelf?
• Problem: From a group of 10 people, how many ways can you select a committee of 4 members?
• Problem: A password consists of 2 letters followed by 3 digits. How many different passwords can be formed if letters and digits can be repeated?
• Problem: How many different ways can you select 3 different fruits from a basket containing 7 different types of fruit?
• Problem: In a class of 15 students, how many ways can you choose a president and a vice-president if the same person cannot hold both positions?

Related Articles:

• Combination and Permutation Practice Questions | Set 2
• Permutation and Combination - Definition, Formulas, Examples