Table of Contents
In mathematics, a combination refers to the selection of items from a larger set. In the case of combinations, the order of selection is irrelevant. This concept is fundamental in many areas of mathematics, especially in probability and statistics. Unlike permutations, where order matters, combinations are all about choosing a specific number of items without worrying about their arrangement. This article will discuss combinations, their definitions, formulas, and examples.
Also Check: Area of a Circle
What are Combinations?
A combination is defined as a selection of items from a larger set, where the sequence in which the items are selected does not matter.
For example, consider a simple set consisting of three elements: P, Q, and R. If we are to select two items from this set, the possible combinations would be PQ, PR, and QR. Notice that PQ is considered the same as QP in combinations because the order of selection is not important.
Formula of Combinations
The combinations formula is a mathematical tool used to determine the number of possible groups or subsets that can be formed by selecting a certain number of objects (r) from a larger set of distinct objects (n), without regard to the order in which the objects are chosen.
The formula for combinations is given by:
Combination Formula = nCr = n!r! (n – r)!
Where: n! (n factorial) is the product of all positive integers up to n.
r! (r factorial) is the product of all positive integers up to r.
(n−r)! is the factorial of the difference between n and r.
Also Check: Vertices, Faces and Edges
Relationship Between Permutation and Combination
A combination is a specific type of permutation where the order of selection is not important. In permutations, different orders of selection are counted as distinct arrangements, whereas, in combinations, they are treated as the same.
For example, in combination AB and BA are the same while in permutation these both are considered different.
As a result, the number of permutations of a set is always greater than or equal to the number of combinations.
The relationship between permutations and combinations is captured by the formula:
nCr = nPr r!
Where, nPr is the number of permutations and nCr is the number of combinations.
Practical Applications of Combinations
Combinations have vast applications in various fields, including statistics, probability, and everyday decision-making processes. Combinations help in calculating the odds in a card game, determining possible outcomes in a survey, or selecting a team from a group of players. Combinations help in analysing possibilities in a systematic way.
Also Check: Area of a Circle
Theorems on Combinations
Theorem: Prove that nPr = r!nCr
Proof:
By the definition of Permutations: nPr = n!(n – r)!
Similarly, by the definition of Combinations: nCr = n!r! (n – r)!
Therefore, the Relation Between Permutations and Combinations:
nCr = n!r! (n – r)!
nCr = nPrr!
nPr = r!nCr
This confirms that the theorem holds true.
Also Check: CUBE
Examples of Combinations
Example 1: Suppose you have a set of four items: Apple, Banana, Cherry, and Durian, and you want to choose any three items from this set to make a milkshake. Find the number of possible combinations.
Ans. According to the combination formula, the number of possible combinations is:
nCr = n!r! (n – r)!
Here, n = 4
r = 3
Therefore, 4C3 = 4!3! (4 – 3)! =4!3! (1)! = 4
Therefore, the possible combinations are four in number. The possible combinations would be ABC, ABD, ACD, and BCD. Where A is apple, B is banana, C is cherry, and D is durian.
Also Check: Average
Example 2: Consider a set of 10 elements, and you want to select 3 elements from this set. Find the number of possible combinations for this situation.
Ans. The number of possible subsets can be calculated using the combination formula:
nCr = n!r! (n – r)!
Here, n = 10 and r = 3
10C3 = 10!3! (10 – 3)! = 10!3! (7)! = 10 9 8 7!3 2 1 (7)! = 10 9 83 2 = 7206 = 3603 = 120
Thus, there are 120 different ways to select 3 items from a set of 10.
Example 3: Consider a set of 5 fruits, and you want to select 3 fruits from this set. Find the number of possible combinations for this situation.
Ans. The number of possible subsets can be calculated using the combination formula:
nCr = n!r! (n – r)!
Here, n = 5 and r = 3
5C3 = 5!3! (5 – 3)! = 5!3! (2)! = 5 4 2 1 = 20 2 = 10
Thus, there are 10 different ways to select 3 items from a set of 5.
Also Check: Circumstance of Circle
Practice Questions on Combinations
- A group of 4 basketball players, A, B, C, and D, needs to form a team of 2 players. In how many ways can the team be selected?
- A committee of 3 people is to be formed from a group of 5 members, P, Q, R, S, and T. How many different committees can be formed?
- There are 6 friends, M, N, O, P, Q, and R. How many ways can a team of 3 people be selected from this group?
- A class has 7 students, X, Y, Z, W, V, U, and T. If the teacher needs to pick a group of 4 students to represent the class in a quiz, how many different groups can be formed?
- In a chess club of 5 members, E, F, G, H, and I, a sub-team of 3 members is to be selected for a tournament. How many possible selections are there?
- A team of 2 swimmers is to be formed from a group of 5 swimmers, K, L, M, N, and O. How many ways can this be done?
- From a set of 6 cards labelled A, B, C, D, E, and F, how many ways can a pair of 2 cards be selected?
- A group of 4 dancers, P, Q, R, and S, need to select 3 dancers for a performance. In how many ways can the selection be made?
Combinations: FAQs
Does Order Matter in Combinations?
No, the order does not matter in combinations. The focus is on the selection of items, not on the arrangement.
What Are Combinations In Numbers?
Combinations are selections where the order of the selected objects does not matter. When selecting r objects out of a given set of n objects, the number of combinations is determined using factorials. The formula for combinations is given by: Combination Formula = nCr = n! / (r! * (n - r)!) This formula calculates the number of different subgroups (combinations) that can be formed from the given larger group of objects.
What is the Formula to calculate the number of Combinations possible?
The formula to calculate the number of possible combinations is given below: Combination Formula = nCr = n! / (r! * (n - r)!)
