Binomial Coefficient Equivalency

In summary, binomial coefficients, also known as combinations, are used in mathematics to calculate the number of ways to choose a subset of objects from a larger set. Binomial coefficient equivalency is a property that states the equivalence of binomial coefficients for two numbers and their complements. It is used in various mathematical concepts and has real-world applications in fields such as statistics and computer science. This property is closely related to symmetry, as it reflects the balance and mirroring in the arrangement of objects being chosen.
  • #1
TranscendArcu
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Find an expression that is identical to [itex]\sum_{k=0}^n \binom{3n}{3k}[/itex]

According to Wolfram, the correct solution to this is: [itex] \frac{1}{3} \left(2(-1)^n + 8^n\right)[/itex]

But I'm not sure which identities of the binomial coefficient I'm supposed to use to prove this. Can anyone give me some direction?

Thanks!
 
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  • #2
Does nobody have any ideas? I was wondering if it were possible to confirm Wolfram's answer via induction, but expanding the resulting binomial coefficients fron the [itex]n-1[/itex] to the [itex]n[/itex] case is proving to be fairly difficult. Any help is appreciated.
 

Related to Binomial Coefficient Equivalency

1. What is a binomial coefficient?

A binomial coefficient, also known as a combination, is a mathematical concept used to calculate the number of ways to choose a subset of objects from a larger set, without regard to the order of the chosen objects.

2. What is binomial coefficient equivalency?

Binomial coefficient equivalency is a property that states that the binomial coefficient of two numbers is equal to the binomial coefficient of their complements. In other words, if n and r are two numbers where n is the total number of objects and r is the number of objects chosen, then the binomial coefficient of (n,r) is equal to the binomial coefficient of (n,n-r).

3. How is binomial coefficient equivalency used in mathematics?

Binomial coefficient equivalency is used in various mathematical concepts such as the binomial theorem, Pascal's triangle, and the combination formula. It allows for the simplification of calculations involving binomial coefficients and provides a way to find the coefficients for larger numbers.

4. What are some real-world applications of binomial coefficient equivalency?

Binomial coefficient equivalency has several applications in fields such as statistics, probability, and computer science. It can be used to calculate the probabilities of certain events, analyze data sets, and optimize algorithms.

5. How is binomial coefficient equivalency related to symmetry?

Binomial coefficient equivalency is closely related to symmetry as it involves the idea of balancing or mirroring. The property states that the binomial coefficient of two numbers is equal, which reflects the symmetry in the arrangement of the objects being chosen. This symmetry is also evident in the symmetric nature of Pascal's triangle, which is a visual representation of binomial coefficients.

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