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

• 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.