binomial coefficient

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 $ax^by^c$ is known as the binomial coefficient $n^b$ or $n^c$ (the two have the same value).
  • These coefficients for varying $n$ and $b$ 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 $a$ in a term $ax^by^c$ of the expanded version is known as a binomial coefficient.
  • where each value $\begin{pmatrix} n \\ k \end{pmatrix}$ 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 $\begin{pmatrix} n \\ k \end{pmatrix}$ 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 $k$ successes in $n$ draws with replacement.

• Combinations

  • The number of $k$-combinations, or $\begin{pmatrix} S \\ k \end{pmatrix}$, 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 $x^k$ term in the polynomial expansion of $(1+x)^n$. $$

• 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 $x+y$.
  • Using the binomial theorem directly, this can be written as:
  • Recall that the binomial coefficients (from the 5th row of Pascal's triangle) are $1, 5, 10, 10, 5, \text{and}\, 1.$ 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 $n$ 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 $N$ much larger than $n$, 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 $C(n, k)$ or $_nC_k$.

• Finding a Specific Term

  • The rth term of the binomial expansion can be found with the equation: ${ \begin{pmatrix} n \\ r-1 \end{pmatrix} }{ a }^{ n-(r-1) }{ b }^{ r-1 }$.
  • 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 $1$ 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 $N$ (the number of trials) and $p$ (the probability of success for each trial) is:
  • $s^2 = Np(1-p)$, where $s^2$ 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.