Newton's Binomial Theorem
Expanding $(2x-y)^4$ by multiplying out brackets by hand is long and it is easy to slip up with the signs. Newton's binomial theorem gives the coefficients of that expansion without multiplying anything out: you just count combinations. It is the same idea that counts paths on a Galton board or genotypes in a population, and it appears whenever something is repeated $n$ times with two possible outcomes.
Factorials and binomial coefficients
The factorial of a natural number, $n!$, is the product of all the natural numbers from $1$ up to $n$ ($0!=1$ by convention). The binomial coefficient $\binom{n}{k}$ counts the number of ways of choosing $k$ elements from a total of $n$, when the order does not matter.
Calculate the value of the following expression using the properties of factorials and binomial coefficients: $\frac{10!}{8! \cdot 2!} + \binom{7}{3}$.
Calculate the value of the following expression using the properties of factorials and binomial coefficients: $\frac{10!}{8! \cdot 2!} + \binom{7}{3}$.
Simplify as far as possible the following algebraic expression involving factorials: $\frac{(n+2)!}{n! \cdot (n+1)}$.
Simplify as far as possible the following algebraic expression involving factorials: $\frac{(n+2)!}{n! \cdot (n+1)}$.
Find the value of the natural number $x$ satisfying the following equation involving binomial coefficients: $\binom{x}{2} = 21$.
Find the value of the natural number $x$ satisfying the following equation involving binomial coefficients: $\binom{x}{2} = 21$.
Pascal's triangle
The binomial coefficients $\binom{n}{0},\binom{n}{1},\dots,\binom{n}{n}$ in row $n$ are exactly the coefficients in the expansion of $(a+b)^n$. Arranged in a triangle, each number is the sum of the two numbers above it: this is Pascal's triangle (or Tartaglia's triangle).
Use Pascal's triangle (or Tartaglia's triangle) to obtain the coefficients of the sixth row and expand the expression $(m + 2n)^6$.
Use Pascal's triangle (or Tartaglia's triangle) to obtain the coefficients of the sixth row and expand the expression $(m + 2n)^6$.
The binomial theorem formula
Newton's binomial theorem expands $(a+b)^n$ without multiplying out the brackets one by one: each term combines a power of $a$ that goes down, a power of $b$ that goes up, and the corresponding coefficient from Pascal's triangle.
Fully expand the binomial $(x + 3)^4$ using the binomial theorem.
Fully expand the binomial $(x + 3)^4$ using the binomial theorem.
Expand the expression $(2a - b)^5$ and state how many terms the final expansion has.
Expand the expression $(2a - b)^5$ and state how many terms the final expansion has.
Expand the binomial with fractions $\left(x + \frac{1}{2}\right)^3$ and simplify each of the resulting terms.
Expand the binomial with fractions $\left(x + \frac{1}{2}\right)^3$ and simplify each of the resulting terms.
The general term: one term without expanding everything
You do not always need the full expansion: if you are only asked for "the fourth term" or "the term in $x^3$", the general term formula gives it directly, without calculating the others.
Write down directly the fourth term in the expansion of the binomial $(x^2 - 2y)^7$, without finding the full expansion.
Write down directly the fourth term in the expansion of the binomial $(x^2 - 2y)^7$, without finding the full expansion.
Find the coefficient of the term containing $x^3$ in the expansion of the binomial $(x - 4)^5$.
Find the coefficient of the term containing $x^3$ in the expansion of the binomial $(x - 4)^5$.
Find the term independent of $x$ (that is, the term that does not contain the variable $x$) in the expansion of the binomial $\left(x^2 - \frac{1}{x}\right)^9$.
Find the term independent of $x$ (that is, the term that does not contain the variable $x$) in the expansion of the binomial $\left(x^2 - \frac{1}{x}\right)^9$.
Radicals, fractions and variable signs in the general term
The general term works just the same when $a$ or $b$ are fractions, roots or carry their own sign; you only need to be careful when simplifying the powers of $x$ whose exponents are added or subtracted.
Determine the middle term in the expansion of the binomial $\left(\sqrt{x} - \frac{2}{x}\right)^6$ and simplify the resulting expression.
Determine the middle term in the expansion of the binomial $\left(\sqrt{x} - \frac{2}{x}\right)^6$ and simplify the resulting expression.
Find the fifth term in the expansion of the binomial $\left(2x^2 - \frac{1}{x}\right)^7$. Simplify the result by combining powers of $x$.
Find the fifth term in the expansion of the binomial $\left(2x^2 - \frac{1}{x}\right)^7$. Simplify the result by combining powers of $x$.
Expand and fully simplify the expression involving radicals $\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^4$. Collect like terms and leave the result without negative…
Expand and fully simplify the expression involving radicals $\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^4$. Collect like terms and leave the result without negative exponents.
Properties of binomial coefficients and applications
Two properties are very useful: the sum of all the coefficients in a row is always $2^n$ (by substituting $a=b=1$), and substituting specific values into the general formula allows you to approximate or model real situations.
Prove that the sum of all binomial coefficients in a given row $n$ satisfies $\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} = 2^n$ using the expansion of the…
Prove that the sum of all binomial coefficients in a given row $n$ satisfies $\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} = 2^n$ using the expansion of the binomial theorem for $(1+1)^n$. Apply this result to determine how many different groups (of any size, including the empty group) can be formed with the $8$ employees of an office.
Calculate the sum of all the coefficients of the polynomial resulting from expanding $(3x - 2y)^5$. Hint: Think about what numerical values you can assign to the…
Calculate the sum of all the coefficients of the polynomial resulting from expanding $(3x - 2y)^5$. Hint: Think about what numerical values you can assign to the variables $x$ and $y$ so that the evaluated expression is exactly equal to the sum of its coefficients.
In economics, the final compound capital accumulated after $5$ years at an annual compound interest rate of $r$ is calculated using the formula $C = C_0(1 + r)^5$. Use…
In economics, the final compound capital accumulated after $5$ years at an annual compound interest rate of $r$ is calculated using the formula $C = C_0(1 + r)^5$. Use the binomial theorem to expand $(1 + r)^5$. If the interest rate is $r = 0,03$, calculate an approximation of the final capital for an initial capital of $1000$ € using only the first three terms of the expansion.
Back to the start: Using the binomial formula, $(2x-y)^4 = (2x)^4 - 4(2x)^3y + 6(2x)^2y^2 - 4(2x)y^3 + y^4 = 16x^4 - 32x^3y + 24x^2y^2 - 8xy^3 + y^4$, solved in a single step without multiplying out brackets by hand.