binomial coefficient

(noun)

A coefficient of any of the terms in the expansion of the binomial power (1+x)n(1+x)^n(1+x)​n​​.

Related Terms

  • Combination
  • k-combination
  • combination
  • binomial coefficients
  • factorial
  • integer
  • binomial

(noun)

A coefficient of any of the terms in the expansion of the binomial power (x+y)n(x+y)^n(x+y)​n​​.

Related Terms

  • Combination
  • k-combination
  • combination
  • binomial coefficients
  • factorial
  • integer
  • binomial

Examples of binomial coefficient in the following topics:

  • 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 axbycax^by^cax​b​​y​c​​ is known as the binomial coefficient nbn^bn​b​​ or ncn^cn​c​​ (the two have the same value).
    • These coefficients for varying nnn and bbb 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
  • 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 aaa in a term axbycax^by^cax​b​​y​c​​ of the expanded version is known as a binomial coefficient.
    • where each value (nk)\begin{pmatrix} n \\ k \end{pmatrix} (​n​k​​) 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
  • 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 (nk)\begin{pmatrix} n \\ k \end{pmatrix}(​n​k​​) is actually an integer.
    • Use factorial notation to find the coefficients of a binomial expansion
  • The Hypergeometric Random Variable

    • This is in contrast to the binomial distribution, which describes the probability of kkk successes in nnn draws with replacement.
  • Combinations

    • The number of kkk-combinations, or (Sk)\begin{pmatrix} S \\ k \end{pmatrix}(​S​k​​), 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 xkx^kx​k​​ term in the polynomial expansion of (1+x)n(1+x)^n(1+x)​n​​.
  • 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 x+yx+yx+y.
    • Using the binomial theorem directly, this can be written as:
    • Recall that the binomial coefficients (from the 5th row of Pascal's triangle) are 1,5,10,10,5,and1.1, 5, 10, 10, 5, \text{and}\, 1.1,5,10,10,5,and1. Using the binomial theorem directly, we have:
    • Connect complex numbers raised to a power to the binomial theorem
  • The Binomial Formula

    • The binomial distribution is a discrete probability distribution of the successes in a sequence of nnn 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 NNN much larger than nnn, 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 C(n,k)C(n, k)C(n,k) or nCk_nC_k​n​​C​k​​.
  • Finding a Specific Term

    • The rth term of the binomial expansion can be found with the equation: (nr−1)an−(r−1)br−1{ \begin{pmatrix} n \\ r-1 \end{pmatrix} }{ a }^{ n-(r-1) }{ b }^{ r-1 }(​n​r−1​​)a​n−(r−1)​​b​r−1​​.
    • 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 111 and they follow Pascal's triangle.
  • 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 NNN (the number of trials) and ppp (the probability of success for each trial) is:
    • s2=Np(1−p)s^2 = Np(1-p)s​2​​=Np(1−p), where s2s^2s​2​​ is the variance of the binomial distribution.
    • Coin flip experiments are a great way to understand the properties of binomial distributions.
  • 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.
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