Permutations and Combinations Level - 1
Q1–10 of 87How many numbers of 3-digits can be formed with the digits 1, 2, 3, 4, 5 (repetition of digits not allowed)?
1. The number of numbers formed would be given by 5 × 4 × 3 (given that the first digit can be filled in 5 ways, the second in 4 ways and the third in 3 ways – MNP rule).
How many numbers between 2000 and 3000 can be formed with the digits 0, 1, 2, 3, 4, 5, 6, 7 (repetition of digits not allowed)?
2. The first digit can only be 2 (1 way), the second digit can be filled in 7 ways, the third in 6 ways and the fourth in 5 ways. A total of 1 × 7 × 6 × 5 = 210 ways.
In how many ways can a person send invitation cards to 6 of his friends if he has four servants to distribute the cards?
3. Each invitation card can be sent in 4 ways. Thus, 4 × 4 × 4 × 4 × 4 × 4 = \(4^{6}\).
In how many ways can 5 prizes be distributed to 8 students if each student can get any number of prizes?
4. In this case since nothing is mentioned about whether the prizes are identical or distinct we can take the prizes to be distinct (the most logical thought given the situation). Thus, each prize can be given in 8 ways — thus a total of \(8^{5}\) ways.
In how many ways can 7 Indians, 5 Pakistanis and 6 Dutch be seated in a row so that all persons of the same nationality sit together?
5. We need to assume that the 7 Indians are 1 person, so also for the 6 Dutch and the 5 Pakistanis. These 3 groups of people can be arranged amongst themselves in 3! ways. Also, within themselves the 7 Indians the 6 Dutch and the 5 Pakistanis can be arranged in 7!, 6! And 5! ways respectively. Thus, the answer is 3! × 7! × 6! × 5!.
There are 5 routes to go from Allahabad to Patna & 4 ways to go from Patna to Kolkata, then how many ways are possible for going from Allahabad to Kolkata via Patna?
6. Use the MNP rule to get the answer as 5 × 4 = 20.
There are 4 qualifying examinations to enter into Oxford University: RAT, BAT, SAT, and PAT. An Engineer cannot go to Oxford University through BAT or SAT. A CA on the other hand can go to the Oxford University through the RAT, BAT & PAT but not through SAT. Further there are 3 ways to become a CA (viz., Foundation, Inter & Final). Find the ratio of number of ways in which an Engineer can make it to Oxford University to the number of ways a CA can make it to Oxford University.
7. An engineer can make it through in 2 ways, while a CA can make it through in 3 ways. Required ratio is 2:3. Option (b) is correct.
How many straight lines can be formed from 8 non-collinear points on the X-Y plane?
8. For a straight line we just need to select 2 points out of the 8 points available. \({}^{8}C_{2}\) would be the number of ways of doing this.
If ${}^{n}C_{3} = {}^{n}C_{8}$, find $n$.
9. Use the property \({}^{n}C_{r}\) = \({}^{n}C_{n-r}\) to see that the two values would be equal at n = 11 since \({}^{11}C_{3}\) = \({}^{11}C_{8}\).
In how many ways can the letters of the word DELHI be arranged?
10. There would be 5! ways of arranging the 5 letters. Thus, 5! = 120 ways.