Proving the Binomial Identity: A Shorter Solution Approach

In summary, the problem is to prove that for all n in N and 0 <= k <= 2^(n-1), C(2^n, k) = sum from j=0 to k of C(2^(n-1), j) * C(2^(n-1), k-j). The attempted solution involves using induction, but the person is looking for a shorter solution. They mention that the RHS is the square of the sum of the term C(2^(n-1), j) and suggest using the fact that 2^n = 2^(n-1) + 2^(n-1) and expanding (1+x)^(2^n) in two different ways. They are seeking
  • #1
MathematicalPhysicist
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problem
prove that:
[tex]\forall n \in N \forall 0<=k<=2^{n-1} (C(2^n,k)=\sum_{j=0}^{k}C(2^{n-1},j)C(2^{n-1},k-j))[/tex]

attempt at solution
induction seems to be too long I am opting for a shorter solution, so the sum that it's wrriten in the rhs is the square of the sum of the term C(2^(n-1),j) but other than that don't know how to procceed.

any advice?

thanks in advance.
 
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  • #2
C is the binomial coefficient.
 
  • #3
so i guess no one here is that good with B.Cs?
 
  • #4
Have you tried using the fact that [tex]2^n=2^{n-1}+2^{n-1}[/tex]
Then expand [tex](1+x)^{2^n}[/tex] in two different ways, equating coefficients of [tex]x^k[/tex]?
 

Related to Proving the Binomial Identity: A Shorter Solution Approach

1. What is the binomial identity?

The binomial identity, also known as the binomial theorem, is a mathematical formula that describes the expansion of a binomial raised to a power. It states that (a + b)^n = Σ(n choose k) * a^(n-k) * b^k, where n is a positive integer, a and b are any real numbers, and (n choose k) denotes the binomial coefficient.

2. What is the significance of proving the binomial identity?

Proving the binomial identity is important in mathematics because it provides a fundamental understanding of how binomials can be expanded and manipulated. It also serves as a basis for many other mathematical concepts and formulas, such as the binomial distribution and Pascal's triangle.

3. What is the traditional approach for proving the binomial identity?

The traditional approach for proving the binomial identity involves using mathematical induction, which is a method of proving a statement for all positive integers by first proving it for the base case (usually n=1) and then showing that if it holds for any integer k, it also holds for the next integer k+1. This process is repeated until the statement is proven for all positive integers.

4. What is the shorter solution approach for proving the binomial identity?

The shorter solution approach for proving the binomial identity involves using the concept of combinatorics, which is the study of counting and arranging objects. This approach uses the fact that the binomial coefficient represents the number of ways to choose a subset of k objects from a set of n objects, and by manipulating this concept, the binomial identity can be proven more efficiently and with less algebraic manipulation.

5. What are the benefits of using the shorter solution approach for proving the binomial identity?

Using the shorter solution approach for proving the binomial identity can save time and effort, as it requires less complex algebraic manipulation compared to the traditional approach. It also provides a deeper understanding of the relationship between combinatorics and the binomial theorem, which can be applied to other mathematical problems. Additionally, this approach can help to develop critical thinking and problem-solving skills.

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