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hxthanh
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Evaluate sum:
$\displaystyle S=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$
$\displaystyle S=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$
I believe that the answer must be \(\displaystyle S = \frac{(-1)^n(6n)!(2n)!}{(4n)!(3n)!n!}\), but I have NO idea how one might prove that.hxthanh said:Evaluate sum:
$\displaystyle S_n=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$
Your result is absolute correct!(Clapping)Opalg said:I believe that the answer must be \(\displaystyle S = \frac{(-1)^n(6n)!(2n)!}{(4n)!(3n)!n!}\), but I have NO idea how one might prove that.
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The sum of binomial coefficients refers to the sum of all the coefficients in a binomial expansion. A binomial expansion is the expansion of a binomial expression raised to a power, such as (a + b)^n. The sum is calculated by adding the coefficients of each term in the expansion.
The sum of binomial coefficients can be calculated using the formula: (1 + 1)^n = 2^n. This formula is derived from the binomial theorem, which states that the sum of the coefficients in a binomial expansion is equal to 2^n, where n is the exponent of the binomial expression.
The sum of binomial coefficients has many applications in mathematics, statistics, and probability. It is used in combinatorics to determine the number of combinations or arrangements of objects. In statistics, it is used to calculate probabilities in binomial distributions. It is also used in algebra and calculus to simplify equations and expressions.
Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers above it. The coefficients in the expansion of (a + b)^n can be represented in Pascal's triangle, with the nth row representing the coefficients of the binomial expansion. The sum of the coefficients in each row of Pascal's triangle is equal to 2^n, which is the sum of binomial coefficients.
Yes, the sum of binomial coefficients can be used to find the coefficient of a specific term in a binomial expansion. This can be done by using the general formula for the coefficients of a binomial expansion, which is nCr = n!/(r!(n-r)!), where n is the exponent and r is the term number. By substituting the values of n and r into this formula, you can find the coefficient of the desired term.