Table of contents
What is factorial?
- Factorial Notation (!)
- n! = n(n – 1) (n – 2) …3.2.1 = Product of n consecutive integers starting from 1.
- 0! = 1
- Factorials of only Natural numbers are defined.
- n! is defined only for n ≥ 0
- n! is not defined for n < 0
- nCr = 1 when n = r.
Combinations (represented by nCr)
Combinations can be defined as the** number of ways in which r things at a time can be SELECTED from amongst n things** available for selection.
The key word here is SELECTION. Please understand here that the order in which the r things are selected has no importance in the counting of combinations.
nCr = Number of combinations (selections) of n things taken r at a time.
where n ≥ r (n is greater than or equal to r).
Some typical situations where selection/combination is used:
- Selection of people for a team, a party, a job, an office etc. (e.g. Selection of a cricket team of 11 from 16 members)
- Selection of a set of objects (like letters, hats, points pants, shirts, etc) from amongst another set available for selection.
In other words any selection in which the order of selection holds no importance is counted by using combinations.
Permutations (represented by nPr)
Permutations can be defined as the number of ways in which r things at a time can be SELECTED & ARRANGED at a time from amongst n things.
The key word here is ARRANGEMENT. Hence please understand here that the order in which the r things are arranged has critical importance in the counting of permutations. In other words permutations can also be referred to as an ORDERED SELECTION.
nPr = number of permutations (arrangements) of n things taken r at a time.
nPr = n!/ (n – r)!; n ≥ r
Some typical situations where ordered selection/ permutations are used:
- Making words and numbers from a set of available letters and digits respectively
- Filling posts with people
- Selection of batting order of a cricket team of 11 from 16 members
- Putting distinct objects/people in distinct places, e.g. making people sit, putting letters in envelopes, finishing order in horse race, etc.)
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1. In how many different ways can five friends sit for a photograph of five chairs in a row?
2. There are 7 candidates for 3 seats, in how many ways the post be filled?
3. How many three-digit numbers can be generated from 1, 2, 3, 4, 5, 6, 7, 8, 9, such that the digits are in ascending order?
4. A square is divided into 9 identical smaller squares. Six identical balls are to be placed in these smaller squares such that each of the three rows gets at least one ball (one ball in one square only). In how many different ways can this be done?
5. Groups each containing 3 boys are to be formed out of 5 boys A,B,C,D and E such that no one group contains both C and D together. What is the maximum number of such different groups?
6. In how many different ways can six players be arranged in a line such that two of them, Asim and Raheem are never together?
7. A person ordered 5 pairs of black socks and some pairs of brown socks. The price of a black pair was thrice that of a brown pair. While preparing the bill, the bill clerk interchanged the number of black and brown pairs by mistake which increased the bill by 100%. What was the number of pairs of brown socks in the original order? (UPSC 2015)
8. There are 5 tasks and 5 persons. Task-l cannot be assigned to either person-l or person-2. Task-2 must be assigned to either person-3 or person-4. Every person is to be assigned one task. In how many ways can the assignment be done? (UPSC 2015)
9. How many numbers are there in all from 6000 to 6999 (both 6000 and 6999 included) having at least one of their digits repeated?
10. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
11. In how many ways can the letters of the word 'LEADER' be arranged?
12. Find the number of permutations of the letters of the word ALLAHABAD.
13. How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?
14. In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?
15. How many motor vehicle registration number plates can be formed with digits 1,2,3,4,5 (no digits being repeated) if it is given that registration number can have 1 to 5 digits?
16. Using all the letters of the word GIFT how many distinct words can be formed?
a. 22 words
b. 24 words
c. 256 words
d. 200 words
17. A person has 4 coins if different denominations. What is the number of different sums of money the person can form?
18. 20 mice were placed in two experimental groups and two control group, with all groups equally large. In how many ways can the mice be placed into three groups?.
19. The letters A, B, C, D and E are arranged in such a way that there are exactly two letters between A and E. How many such arrangements are possible? (UPSC 2022)
20. There is a numeric lock which has a 3-digit PIN. The PIN contains digits 1 to 7. There is no repetition of digits. The digits in the PIN from left to right are in decreasing order. Any two digits in the PIN differ by at least 2. How many maximum attempts does one need to find out the PIN with certainty? (UPSC 2022)
21. There are 9 cups placed on a table arranged in equal number of rows and columns out of which 6 cups contain coffee and 3 cups contain tea. In how many ways can they be arranged so that each row should contain at least one cup of coffee?
22. In how many of the distinct permutations of the letters in MISSISSIPPI do the four Ss not come together?
23. There are 45 games in total in a competition. Many teams took part in the competition and each of them must play one with the other teams. In total how many teams took part in the competition?