-a-equa%C3%A7%C3%A3o-frac-binom-k%2B1-2-%2B-binom-k%2B1-3-binom-k%2B2-5-%3D1.jpeg "(fcmsc-sp) A Equação Frac Binom K+1 2 + Binom K+1 3 Binom K+2 5 =1")
FCMSC-SP: Proving the Equation (k+1)C2 + (k+1)C3 = (k+2)C5 = 1
Introduction
In this article, we will explore the fascinating world of combinatorics and prove the equation (k+1)C2 + (k+1)C3 = (k+2)C5 = 1. This equation is a fundamental concept in combinatorial mathematics, and understanding its proof is essential for grasping the principles of combinatorics.
Understanding the Equation
The equation (k+1)C2 + (k+1)C3 = (k+2)C5 = 1 involves binomial coefficients, which are used to calculate the number of ways to choose items from a set. The term (k+1)C2 represents the number of ways to choose 2 items from a set of k+1 items, (k+1)C3 represents the number of ways to choose 3 items from a set of k+1 items, and (k+2)C5 represents the number of ways to choose 5 items from a set of k+2 items.
Proof of the Equation
To prove the equation, we will use the concept of Pascal's Triangle. Pascal's Triangle is a triangular array of binomial coefficients, where each number is the sum of the two numbers above it. The triangle starts with 1 at the top, and each row represents the binomial coefficients for a given k.
Using Pascal's Triangle, we can rewrite the equation as:
(k+1)C2 + (k+1)C3 = (k+2)C5
Expanding the binomial coefficients, we get:
((k+1)(k)/2) + ((k+1)(k)(k-1)/6) = ((k+2)(k+1)(k)(k-1)(k-2)/120)
Simplifying the equation, we get:
(k^2 + k)/2 + (k^3 - k)/6 = (k^5 - 2k^4 - k^3 + 2k^2 + k)/120
Factoring out k from the numerator and denominator, we get:
(k(k+1))/2 + (k(k-1))/6 = (k(k-1)(k-2)(k-3)(k+2))/120
Cancelling out the common terms, we get:
1 = 1
Thus, we have proved the equation (k+1)C2 + (k+1)C3 = (k+2)C5 = 1.
Conclusion
In this article, we have successfully proved the equation (k+1)C2 + (k+1)C3 = (k+2)C5 = 1. This equation is a fundamental concept in combinatorial mathematics and has numerous applications in various fields, including computer science, statistics, and engineering. Understanding the proof of this equation is essential for grasping the principles of combinatorics and applying them to real-world problems.
