A combination is a way of choosing items from a set (unlike a permutation) when the order of selection doesn't matter. In smaller cases, it's possible to count the number of combinations. A 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 elements, then there is only one combination, {a, b, c} which means 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.
Formula
The number of combinations when ‘r’ elements are selected out of a complete set of ‘n’ elements is denoted by nCr and is given by the formula:
Example: Let n = 4 (E, F, G, H) and r = 2 (consisting of all the combinations of size 2).Â
nCr = 4C2
⇒ 4C2 = 4!/((4-2)!×2!)Â
⇒ 4C2 = (4 × 3 × 2 × 1)/(2 × 1 × 2 × 1) = 6
The six combinations are EF, EG, EH, FG, FH, and GH.
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 objects 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!
Note: The key difference between Permutation and Combination:
- 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.
How to Calculate the Probability of Combinations?
Probability of Combinations can be easily understood with the help of the examples given below:
Example:
- How many ways can we choose 3 candies from 7 different candies?
- • In how many ways can we choose 2 letters from the word ‘POWER’?
- How many ways can we select 4 digits from the digits 2, 3, 5, 6, 7, and 9?
Solution:
For 1: We are choosing 3 candies out of 7, where order does not matter.
\binom{7}{3}= \frac{7!}{3! \cdot 4!} = 35 For 2: We are choosing 2 letters from 5 distinct letters in ‘POWER’.
\binom{5}{2} = \frac{5!}{2! \cdot 3!} = 10 For 3: We are selecting 4 digits out of 6 distinct digits.
\binom{6}{4} = \frac{6!}{4! \cdot 2!} = 15
Handshaking Problem
Handshaking problem is one of the most interesting problems 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-BÂ | 1 handshake |
| Three People | A-B A-CÂ B-C | 3 handshake |
| Four People | A-B A-C A-D B-C B-D C-D | Â 6 handshakes |
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
