binomial coefficient

(noun)

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

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 $(x+y)^n$.

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 $ax^by^c$ 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
  • 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
  • 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.
  • Multiplying Algebraic Expressions

    • When you multiply monomials, you multiply their integer coefficients together and, if they contain any of the same variables, add the exponents on those variables together.
    • Multiplying two binomials is less straightforward; however, there is a method that makes the process fairly convenient.
    • Outer (the "outside" terms are multiplied—i.e., the first term of the first binomial with the second term of the second)
    • Inner (the "inside" terms are multiplied—i.e., the second term of the first binomial with the first term of the second)
    • Remember that any negative sign on a term in a binomial should also be included in the multiplication of that term.
  • Sums, Differences, Products, and Quotients

    • A monomial equations has one term; a binomial has two terms; a trinomial has three terms.
    • Multiplying binomials and trinomials is more complicated, and follows the FOIL method.
    • FOIL is a mnemonic for the standard method of multiplying two binomials; the method may be referred to as the FOIL method.
    • Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
    • Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
  • Graphing Quadratic Equations In Standard Form

    • Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function's graph.
    • The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex.
    • If the coefficient $a>0$, the parabola opens upward, and if the coefficient $a<0$, the parabola opens downward.
    • The coefficients $b$ and $a$ together control the axis of symmetry of the parabola and the $x$-coordinate of the vertex.
    • The coefficient $c$ controls the height of the parabola.
  • Factoring Perfect Square Trinomials

    • When a trinomial is a perfect square, it can be factored into two equal binomials.
    • Note that if a binomial of the form $a+b$ is squared, the result has the following form: $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.$ So both the first and last term are squares, and the middle term has factors of $2, $ $a$, and $b,$ where the latter are the square roots of the first and last term respectively.
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