Derivation of nCr Formula
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)!]
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.
Table of Content
- What is Combination?
- What is nCr Formula?
- Properties of nCr
- Derivation of nCr Formula
- nPr and nCr Formula