In this article, we will discuss the basic concepts of Permutation & Combination and formulas required for solving problems on the same. This topic is covered in all competitive exams, so you cannot afford to take the risk of avoiding it. After reading this article, you will become familiar with the basics concepts of permutation and combination.
Permutation and Combination formula
Solved Permutation and Combination Problems
Example 1: How many four-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6 (Repetition of digits not allowed)?
Solution: Thousand's place can be filled in 6 ways. Hundred's place can be filled in 5 ways. Ten's place can be filled in 4 ways. Unit's place can be filled in 3 ways. So, using the Fundamental Principle of Counting, we get the answer as 6 × 5 × 4 × 3 = 360. Or using formula of Permutations, we need to arrange 4 digits out of total 6 digits. This can be done in ^{6}P_{4} = 360 ways.
Example 2: A person has 6 friends to be invited for dinner through invitation cards, and he has 3 servants. In how many ways can he extend the invitation card?
Solution: We can see that the 1^{st} friend has 3 options to receive the card, i.e. either from 1^{st} servant or 2^{nd} or 3^{rd}. Similarly 2nd friend also has 3 options to receive the card, i.e. either from 1^{st} servant or 2^{nd} or 3^{rd}. So we can say that each of the 6 friends has 3 options to receive the card. Hence the answer would be 3 × 3 × 3 × 3 × 3 × 3 = 3^{6} = 729 ways.
Example 3: There are 10 questions in an exam. In how many ways can a person attempt at least one question?
Solution: A person can attempt 1 question or 2 questions or .....till all 10 questions. One question out of ten questions can be attempted in ^{10}C_{1} = 10 ways. Similarly two questions out of ten questions can be attempted in ^{10}C_{2} = 45 ways. Going ahead by the same logic, all ten questions can be attempted in ^{10}C_{10} = 1 way. Hence the total number of ways = 10 + 45 + 120 +.....10 + 1 = 1023 ways (Using the formula of Combination).
Alternate Method:
Or some logic can be applied: Every question has 2 options, either it is attempted or not. Going ahead with this logic, since there are 10 questions, and each question has 2 options, so total number of cases = 2^{10}= 1024. But this count includes one case in which no question is attempted. This is the violation of the information given. So this case needs to be subtracted. Hence the total number of cases would be 1024 - 1 = 1023.Permutation and Combination: Key Learnings
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