binomial coefficient
A coefficient of any of the terms in the expansion of the binomial power
A coefficient of any of the terms in the expansion of the binomial power
Examples of binomial coefficient in the following topics:
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Total Number of Subsets
- The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.
- The binomial theorem describes the algebraic expansion of powers of a binomial.
- The coefficient a in the term of is known as the binomial coefficient or (the two have the same value).
- These coefficients for varying and can be arranged to form Pascal's triangle.
- Employ the Binomial Theorem to find the total number of subsets that can be made from n elements
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Binomial Expansions and Pascal's Triangle
- The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
- Any coefficient in a term of the expanded version is known as a binomial coefficient.
- where each value is a specific positive integer known as binomial coefficient.
- For a binomial expansion with a relatively small exponent, this can be a straightforward way to determine the coefficients.
- Use the Binomial Formula and Pascal's Triangle to expand a binomial raised to a power and find the coefficients of a binomial expansion
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Binomial Expansion and Factorial Notation
- The binomial theorem describes the algebraic expansion of powers of a binomial.
- The coefficients that appear in the binomial expansion are called binomial coefficients.
- The coefficient of a term $x^{n−k}y^k$ in a binomial expansion can be calculated using the combination formula.
- Note that although this formula involves a fraction, the binomial coefficient is actually an integer.
- Use factorial notation to find the coefficients of a binomial expansion
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The Hypergeometric Random Variable
- This is in contrast to the binomial distribution, which describes the probability of successes in draws with replacement.
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Combinations
- The number of -combinations, or , is also known as the binomial coefficient, because it occurs as a coefficient in the binomial formula.
- The binomial coefficient is the coefficient of the term in the polynomial expansion of .
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Complex Numbers and the Binomial Theorem
- Powers of complex numbers can be computed with the the help of the binomial theorem.
- Recall the binomial theorem, which tells how to compute powers of a binomial like .
- Using the binomial theorem directly, this can be written as:
- Recall that the binomial coefficients (from the 5th row of Pascal's triangle) are Using the binomial theorem directly, we have:
- Connect complex numbers raised to a power to the binomial theorem
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The Binomial Formula
- The binomial distribution is a discrete probability distribution of the successes in a sequence of independent yes/no experiments.
- This makes Table 1 an example of a binomial distribution.
- The binomial distribution is the basis for the popular binomial test of statistical significance.
- However, for much larger than , the binomial distribution is a good approximation, and widely used.
- Is the binomial coefficient (hence the name of the distribution) "n choose k," also denoted or .
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Finding a Specific Term
- The rth term of the binomial expansion can be found with the equation: .
- You might multiply each binomial out to identify the coefficients, or you might use Pascal's triangle.
- Let's go through a few expansions of binomials, in order to consider any patterns that are present in the terms.
- The coefficients of the first and last terms are both and they follow Pascal's triangle.
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Mean, Variance, and Standard Deviation of the Binomial Distribution
- In this section, we'll examine the mean, variance, and standard deviation of the binomial distribution.
- The easiest way to understand the mean, variance, and standard deviation of the binomial distribution is to use a real life example.
- In general, the mean of a binomial distribution with parameters (the number of trials) and (the probability of success for each trial) is:
- , where is the variance of the binomial distribution.
- Coin flip experiments are a great way to understand the properties of binomial distributions.
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Additional Properties of the Binomial Distribution
- In this section, we'll look at the median, mode, and covariance of the binomial distribution.
- There are also conditional binomials.
- The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non-identical Bernoulli trials Bern(pi).
- This formula is for calculating the mode of a binomial distribution.
- This summarizes how to find the mode of a binomial distribution.