A combination is a way of selecting objects from a group where the order of selection does not matter.
Example: If you have 5 students and need to choose 2 students for a team, choosing A and B is the same as choosing B and A.
Solved Examples
Example 1: In how many ways can 6 boys 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 be two people present at the party who have 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
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?
