Combination is a way of choosing items from a set, (unlike permutations) when the order of selection doesn't matter. In smaller cases, it's possible to count the number of combinations. Combination refers to the mixture of n things taken k at a time without repetition.
Example: For set S = {a, b, c}, the possible combinations of choosing 2 elements are, {a, b}, {a, c}, {b, c}. If we choose 3 items, then there is only one combination {a, b, c} which is we pick all three.
Some other examples of Combination in Real-Life are:
- Team Selection: Choosing 5 players from a pool of 12 for a sports team.
- Menu Selection: Picking a combination of dishes (starter, main course, dessert) from a menu.
- Lottery Numbers: Selecting a set of lottery numbers where the order doesn’t matter.
Combination is the choice of selecting r things from a group of n things without replacement and where the order of selection is not important. The number of combinations when ‘r’ elements are selected out of a complete set of ‘n’ elements is denoted by nCr
Example: Let n = 4 (E, F, G, H) and r = 2 (consisting of all the combinations of size 2).
nCr = 4C2
⇒ 4C2 = 4!/((4-2)!×2!)
⇒ 4C2 = (4 × 3 × 2 × 1)/(2 × 1 × 2 × 1) = 6
The six combinations are EF, EG, EH, FG, FH, and GH.
Read More about Combinations Formula.
Relationship between Permutation and Combination
Permutation and combination have a lot of similarities but they also have some striking differences. For n different objects, we have to make r unique selections from this group of n objects.
The number of permutations of size r from n object is nPr here the order, of selection is not important so each selection is counted r! times. So the number of unique selections is nPr / r! We know that a unique selection of r things from the total of n things is called a combination (nCr). Thus,
nCr = nPr / r!
Read More about Principle of Couting.
Difference Between Permutation and Combination
The key difference between Permutation and Combination is:
- Permutation considers the order of arrangement, meaning the arrangement of items matters. For example, arranging letters to form words or seating people in a row.
- Combination ignores the order, meaning only the selection of items matters, not their arrangement. For example, selecting a committee from a pool of candidates or choosing lottery numbers.
Read detailed difference between Permutation and Combination.
How to Calculate Probability of Combinations?
Probability of Combinations can be easily understood with the help of the examples given below:
Example:
- How many ways are possible to distribute 7 different candies to 3 people where each gets only 1 candy?
- In how many ways can the letters of the word ‘POWER’ be arranged?
- How many six-digit numbers can be formed with digits 2,3,5, 6, 7, and 9 and with distinct digits?
Solution:
For 1:
For the first people, we can choose any of the candy among the 7 candies available. Similarly, for the 2nd person we are left with 6 choices and for the 3rd, we will be having 5 choices.
So, the number of ways of distributing candies = 7 × 6 × 5 = 210 ways
For 2:
Letters of the word ‘POWER’ can be arranged in 5! ways i.e.
5 × 4 × 3 × 2 × 1 ways = 120 ways.
For 3:
Number of distinct ways of forming 6-digit numbers with different digits is
6! = 6 × 5 × 4 × 3 × 2 × 1 ways
= 720 ways.
Also, to learn about probability and combinations, read: How To Calculate Probability using Combination?
What is Handshaking Problem?
Handshaking problem is one of the most interesting problem related to combinations. It is used to find that in a room full of people how many handshakes are required for everybody to shake everybody else's hand exactly once?
Example: The table given below tells us about the minimum number of handshakes required for various groups of people.
| Number of People | Possible Combinations | Minimum Handshake required |
|---|
| Two People | A-B | 1 handshake |
| Three People | A-B A-C B-C | 3 handshake |
| Four People | A-B A-C A-D B-C B-D C-D | 6 handshake |
When there are few people, handshakes can be counted individually. However, with thousands of people in a hall, counting each handshake becomes impractical, which is where combinations come into play.
Handshaking Combination
It means the total number of people in a room doing the handshake with each other. With the help of combination formulas, it can easily be calculated. The formula for calculating the handshakes when there are n people available is given by,
- Total Number of Handshakes = n × (n - 1)/2
- Total Number of Handshakes = nC2
Article Related to Combinations
Examples on Combinations
Example 1: In how many ways 6 boys can be arranged in a queue such that
a) Two particular boys of them are always together
b) Two particular boys of them are never together
Solution:
a) If two boys are always together, then they will be treated as one entity. Hence we can be arranged 5 boys in 5! ways. Also, two boys can arrange themselves in 2 different ways.
Therefore required arrangement = 5! × 2 = 120 × 2 = 240 ways.
b) Total number of permutations among 6 numbers is given by = 6! = 720.
In 240 cases 2 boys are always together.
Thus, for two boys who are never together no of ways will be = 720 – 240 = 480 ways.
Example 2: In a room of n people, how many handshakes are possible?
Solution:
To see the people present, and consider one person at a time. The first person will shake hands with n - 1 other people. The next person will shake hands with n-2 other people, not counting the first person again. Following this, it will give us a total number of
(n - 1) + (n - 2) + ... + 2 + 1
= n(n - 1)/ 2 handshakes.
Example 3: Another popular handshake problem starts out similarly with n>1 people at a party. Not being possible to shake hands with yourself, and not counting several times handshakes with the same person, the problem is to show that there will always present two people at the party, who had shaken hands the same number of times in the party.
Solution:
Solution to this problem starts by using Dirichlet's box principle. If there exists a person at the party, who has shaken hands zero times, then every person which is there at the party has shaken hands with at most n-2 other people at the party.
There are n-1 possible handshakes (from 0 to n-2), among n people there must be two who have shaken hands the same number of times. If there are zero persons, who has shaken hands zero times this means that all of the party guests have shaken hands at least once.
This also amounts to n-1 possible handshakes (from 1 to n-1).
Example 4: In the function, if every person shakes hands with every other in the party and there exists a total of 28 handshakes at the party, find the number of persons who were present in the function.
Solution:
Suppose there are n persons present at a party and every person shakes hands with every other person.
Then, total number of handshakes = nC2 = n(n - 1)/2
⇒ n(n - 1)/2 = 28
⇒ n(n - 1) = 28 × 2
⇒ n(n - 1) = 56
⇒ n = 8
Practice Questions on Combinations in Maths
Q1. A classroom has 20 students, and a committee of 4 students needs to be formed to organize an upcoming event. In how many ways can this committee be chosen?
Q2. You have 5 different books on mathematics and want to select 3 to place on your desk for quick reference. In how many ways can you choose which 3 books to place on the desk?
Q3. A fruit shop offers baskets that can contain any combination of 3 different fruits from their selection of 5 different fruits (apple, banana, cherry, date, and elderberry). How many different fruit baskets can be made?
Q4. A lottery game requires you to choose 6 numbers out of a possible 49. How many different combinations of numbers can you choose?
Q5. From a pool of 12 jurors, a jury of 6 needs to be selected for a trial. In how many different ways can the jury be formed?
Combination in Mathematics
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