nCr| Combinations Formula
Last Updated :
13 Aug, 2024
nCr represents the number of ways to choose r items from a set of n distinct items without regard to the order of selection. Combinations Formula is one of the countless formulas in the world of mathematics, which plays a pivotal role in solving problems.
nCr is expressed as n!/r!(n-r)!
They are widely used in probability and statistics to calculate the possible outcomes of events. They also have many applications in real-life situations such as forming teams, choosing passwords, arranging books, etc. In this article, we will explore the nCr formula in detail, discussing its importance, and applications and providing clarity through solved problems.
What is Combination?
Combinations are a fundamental concept in combinatorics, a branch of mathematics that deals with counting, and selecting objects without regard to their specific order. In simple words, combinations are used to determine the number of ways to select a specific number of items from a larger set, without considering the order in which the items are chosen.
The number of combinations of "n" items taken "r" at a time is denoted as "C(n, r)" or "n choose r" or " nCr ".
nCr represents "n choose r," a concept in combinatorics that calculates the number of ways to select a group of items from a larger set without considering the order of selection. It is denoted mathematically as:
nCr = n! / (r!(n - r)!)
Where,
- "n" is the total number of items in the set,
- "r" is the number of items to be chosen, and
- "!" denotes factorial, which is the product of all positive integers from 1 to the given number.
Note: nCr Formula is also called Combination Formula.
Properties of nCr
Some of the common properties of nCr are:
- nCr is the same as nCn-r, which means that choosing r items out of n items is equivalent to choosing (n-r) items out of n items.
- nCr is a natural number, which means that it is always a positive integer. For example, 4C2 is 6, which is a natural number.
- nCr follows the binomial theorem, which means that it can be used to find the coefficients of the expansion of (x + y)n. For example, the coefficients of (x + y)4 are 4C0, 4C1, 4C2, 4C3, and 4C4, which are 1, 4, 6, 4, and 1 respectively.
- nC1 = n, which means there are 'n' ways of choosing one object from 'n' objects.
- nCx = nCy implies either x = y or x + y = n, which means the number of ways of choosing x objects from n objects is equal to the number of ways of choosing y objects from n objects only if x and y are equal or complementary (sum up to n).
The nCr formula is a way of counting how many different combinations of r items can be chosen from a set of n items. To derive this formula, we can use nPr Formula as follows:
Derivation Using nPr and nCr Relation
nPr = nCr × r!
Using this relation, we can derive the nCr formula from the nPr formula as follows:
- Start with the formula for permutations nPr=n! / (n-r)!
- Substitute nPr with C(n, r) × r ! using the relation above
- Solve for nCr by dividing both sides by r!
This gives us:
nPr = nCr × r !
⇒ n! / (n-r)! = nCr * r !
⇒ [n! / (n-r)!] / r ! = nCr
⇒ nCr = n! / [r! × (n-r)!]
The nPr and nCr formulas are used to calculate the number of ways to arrange or select objects from a given set of objects. The difference between them is that nPr considers the order of the objects, while nCr does not.
| Formula | Interpretation | Formula Expression |
|---|
| nPr | Permutations of "n" objects taken "r" at a time. | nPr = n! / (n - r)! |
| nCr | Combinations of "n" objects taken "r" at a time. | nCr = n! / (r! × (n - r)!) |
Where,
- n is the total number of objects,
- r is number of objects taken at a time, and
- Factorial i.e., n! is the product of all positive integers from 1 to "n."
- i.e., n! = n × (n - 1) × (n - 2) × . . . × 2 × 1
For example, if we have 3 letters A, B, and C, and we want to arrange them in different ways, we can use the nPr formula. Using the nPr formula we get the answer 6 for this arrangement. This means that there are 6 ways to arrange the 3 letters: ABC, ACB, BAC, BCA, CAB, and CBA.
However, if we want to select 2 letters out of the 3 letters, without caring about the order, we can use the nCr formula. Using the formula we get 3 as result, this means that there are 3 ways to select 2 letters out of the 3 letters: AB, AC, and BC.
There are various application of nCr formula are:
- Probability: In probability theory, nCr is used to calculate the probability of certain events. For example, in a lottery, you can use nCr to calculate the probability of winning by choosing the winning numbers out of the total number of possible combinations.
- Binomial Coefficients: nCr appears in the binomial coefficient formula, which is used to expand binomial expressions. For example, in (a + b)n, the coefficients of each term can be calculated using nCr.
- Counting Subsets: You can use nCr to count the number of distinct subsets of a set with n elements. For instance, if you have a set of 5 elements, 5C0, 5C1, 5C2, 5C3, 5C4, and 5C5 represent the number of subsets with 0, 1, 2, 3, 4, and 5 elements, respectively.
Also, Check
Solved Problems on nCr (Combinations Formula)
Problem 1: You are at an ice cream parlor, and they offer 10 different flavors of ice cream. You can choose 3 scoops of ice cream. How many different combinations of ice cream can you order?
Solution:
This is a combination problem. You want to find 10 choose 3 (10C3).
10C3 = 10! / (3!(10-3)!) = 120 different combinations.
So, there are 120 different ways to order 3 scoops of ice cream from 10 flavors.
Problem 2: In a school with 30 students, there are 5 positions available on the student council: president, vice-president, secretary, treasurer, and historian. How many different ways can the positions be filled?
Solution:
This is a permutation problem because the order of election to the positions matters.
For the president, there are 30 choices.
For the vice-president, there are 29 choices (since one person is already president).
For the secretary, there are 28 choices (after president and vice-president are selected).
For the treasurer, there are 27 choices.
For the historian, there are 26 choices.
Now, multiply these choices together to get the total number of ways to fill the positions:
30 × 29 × 28 × 27 × 26 = 17,956,800 different ways to fill the positions.
So, there are 17,956,800 different ways to elect the student council.
Problem 1: In a lottery game, you need to choose 6 numbers from a pool of 49. How many different combinations of numbers can you choose?
Problem 2: A restaurant has a menu with 15 different main dishes, 8 different appetizers, and 10 different desserts. If a customer wants to order one main dish, one appetizer, and one dessert, how many different meal combinations are possible?
Conclusion
The nCr is representation of combinations formula which is a fundamental tool in combinatorics, essential for determining the number of ways to choose r items from a set of n distinct items without regard to order. By leveraging the combination formula n!/r!⋅(n−r)!, we can efficiently calculate the number of possible subsets or combinations of r items out of n available items in various applications, from probability theory to statistical analysis and beyond.
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