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- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_2e_(OpenStax)/11%3A_Sequences_Probability_and_Counting_Theory/11.07%3A_Binomial_TheoremA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consumi...A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find (x+y)n without multiplying the binomial by itself n times.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/11%3A_Sequences_Probability_and_Counting_Theory/11.06%3A_Binomial_TheoremA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consumi...A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find (x+y)n without multiplying the binomial by itself n times.
- https://math.libretexts.org/Courses/Truckee_Meadows_Community_College/TMCC%3A_Precalculus_I_and_II/Under_Construction_test2_11%3A_Sequences_Probability_and_Counting_Theory/Under_Construction_test2_11%3A_Sequences_Probability_and_Counting_Theory_11.6%3A_Binomial_TheoremA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consumi...A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find (x+y)n without multiplying the binomial by itself n times.
- https://math.libretexts.org/Bookshelves/Algebra/Advanced_Algebra/09%3A_Sequences_Series_and_the_Binomial_Theorem/9.04%3A_Binomial_TheoremThe binomial theorem provides a method of expanding binomials raised to powers without directly multiplying each factor.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/08%3A_Combinatorics/8.05%3A_The_Binomial_TheoremA binomial is a polynomial with exactly two terms. The binomial theorem gives a formula for expanding (x+y)ⁿ for any positive integer n .
- https://math.libretexts.org/Courses/Palo_Alto_College/College_Algebra/06%3A_Sequences_Probability_and_Counting_Theory/6.06%3A_Binomial_TheoremA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consumi...A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find (x+y)n without multiplying the binomial by itself n times.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/1%3A_Counting/1.2%3A_Binomial_CoefficientsHere are some apparently different discrete objects we can count: subsets, bit strings, lattice paths, and binomial coefficients. We will give an example of each type of counting problem (and say what...Here are some apparently different discrete objects we can count: subsets, bit strings, lattice paths, and binomial coefficients. We will give an example of each type of counting problem (and say what these things even are). As we will see, these counting problems are surprisingly similar.
- https://math.libretexts.org/Bookshelves/Algebra/College_Algebra_1e_(OpenStax)/09%3A_Sequences_Probability_and_Counting_Theory/9.07%3A_Binomial_TheoremA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consumi...A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find (x+y)n without multiplying the binomial by itself n times.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/01%3A_Fundamentals/1.04%3A_Binomial_Coefficients\[\eqalign{ \sum_{i=0}^{n-1}{n-1\choose i}&x^{n-i}y^i+ \sum_{i=0}^{n-1} {n-1\choose i}x^{n-1-i}y^{i+1}\cr &=\sum_{i=0}^{n-1}{n-1\choose i}x^{n-i}y^i+ \sum_{i=1}^{n} {n-1\choose i-1}x^{n-i}y^{i}\cr &={...n−1∑i=0(n−1i)xn−iyi+n−1∑i=0(n−1i)xn−1−iyi+1=n−1∑i=0(n−1i)xn−iyi+n∑i=1(n−1i−1)xn−iyi=(n−10)xn+n−1∑i=1(n−1i)xn−iyi+n−1∑i=1(n−1i−1)xn−iyi+(n−1n−1)yn=(n−10)xn+n−1∑i=1((n−1i)+(n−1i−1))xn−iyi+(n−1n−1)yn.
- https://math.libretexts.org/Courses/Chabot_College/Chabot_College_College_Algebra_for_BSTEM/09%3A_Sequences_Probability_and_Counting_Theory/9.06%3A_Binomial_TheoremA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consumi...A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find (x+y)n without multiplying the binomial by itself n times.
- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/11%3A_Sequences_Probability_and_Counting_Theory/11.07%3A_Binomial_TheoremA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consumi...A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find (x+y)n without multiplying the binomial by itself n times.
