nCr Formula

nCr Formula is one of the countless formulas in the world of mathematics, which plays a pivotal role in solving problems and gaining a deeper understanding of the subject. nCr formula deals with combinations and as we know, Combinations are an integral part of combinatorics, the branch of mathematics that focuses on counting and arranging objects.

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 “ⁿC_r“.

What is ⁿC_r Formula?

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:

ⁿC_r = 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: ⁿC_r Formula is also called Combination Formula.

Properties of ⁿC_r

Some of the common properties of ⁿC_r are:

• ⁿC_r is the same as ⁿC_n-r, which means that choosing r items out of n items is equivalent to choosing (n-r) items out of n items.
• ⁿC_r is a natural number, which means that it is always a positive integer. For example, ⁴C₂ is 6, which is a natural number.
• ⁿC_r 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.
• ⁿC_x = ⁿC_y 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).

Derivation of ⁿC_r Formula

The ⁿC_r 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 ⁿP_r Formula as follows:

Derivation Using ⁿP_r and ⁿC_r Relation

ⁿP_r = ⁿC_r × r!

Using this relation, we can derive the nCr formula from the nPr formula as follows:

• Start with the formula for permutations ⁿP_r=n! / (n-r)!
• Substitute ⁿP_r with C(n, r) × r ! using the relation above
• Solve for ⁿC_r 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)!]

ⁿP_r and ⁿC_r Formula

The ⁿP_r and ⁿC_r 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 ⁿP_r considers the order of the objects, while ⁿC_r 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 ⁿP_r formula. Using the ⁿP_r 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.

Applications of ⁿC_r Formula

There are various application of ⁿC_r formula are:

• Probability: In probability theory, ⁿC_r is used to calculate the probability of certain events. For example, in a lottery, you can use ⁿC_r to calculate the probability of winning by choosing the winning numbers out of the total number of possible combinations.
• Binomial Coefficients: ⁿC_r 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 ⁿC_r.
• 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.

Solved Problems

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.

Practice Problems on nCr Formula

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?

ⁿC_r Formula: FAQs

1. What is the Value of ⁰C_n?

⁰C_n is defined as 1. This may seem counterintuitive, but it is based on combinatorial principles.

2. How is ⁿC_r used in the Binomial Theorem?

The binomial theorem relies on the expansion of binomial expressions, which includes coefficients represented by ⁿC_r.

3. Can ⁿC_r be Calculated using a Calculator?

Yes, modern calculators often have a function for calculating ⁿC_r directly.

4. What is the Relationship between ⁿC_r and ⁿP_r?

ⁿC_r calculates combinations, which disregard order, while ⁿP_r calculates permutations, which consider order. The relationship is expressed as ⁿP_r = ⁿC_r × r!

5. Are there any Limitations to using the ⁿC_r Formula?

The main limitation is that it can become computationally intensive for large values of n and r. In such cases, more efficient algorithms are preferred.