The Binomial Theorem
Emma makes 70% of her free throws. She takes 5 shots in practice. What is the probability that she makes exactly 3? Multiplying out $(0.7 + 0.3)^5$ by hand would take forever. The binomial theorem gives every term of $(a+b)^n$ directly by counting combinations. That same idea gives the answer to Emma's question, which you'll compute at the end of this page.
Factorials and Binomial Coefficients
The factorial of a whole number $n$ is the product of all the whole numbers from $n$ down to $1$. We write it $n!$. By definition, $0! = 1$.
The binomial coefficient $\binom{n}{k}$ counts the ways to choose $k$ items from $n$ items when order doesn't matter. It is the same number as $_nC_k$ on your calculator, and you read it "$n$ choose $k$."
Why $\binom{n}{k} = \binom{n}{n-k}$? Choosing the $k$ players who make the team is the same as choosing the $n-k$ players who stay home.
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 the following algebraic expression with factorials completely: $\frac{(n+2)!}{n! \cdot (n+1)}$.
Simplify the following algebraic expression with factorials completely: $\frac{(n+2)!}{n! \cdot (n+1)}$.
Pascal's Triangle
The binomial coefficients $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$ are the numbers in row $n$ of Pascal's triangle. The top row is row $n = 0$. Every row starts and ends with $1$, and every other number is the sum of the two numbers above it. Later you'll see that row $n$ gives the coefficients of $(a+b)^n$.
Use Pascal's triangle to obtain the coefficients of the sixth row and expand the expression $(m + 2n)^6$.
Use Pascal's triangle to obtain the coefficients of the sixth row and expand the expression $(m + 2n)^6$.
The Binomial Theorem
To expand $(a+b)^n$, combine three things in each term: a coefficient from Pascal's triangle, a power of $a$ that goes down, and a power of $b$ that goes up.
Completely expand the binomial $(x + 3)^4$ using the binomial theorem.
Completely 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.
The General Term
Sometimes you only need one term, such as "the 4th term" or "the term with $x^3$." The general term gives it directly, so you never expand the rest.
Write down the fourth term of the expansion of the binomial $(x^2 - 2y)^7$ directly, without calculating the whole expansion.
Write down the fourth term of the expansion of the binomial $(x^2 - 2y)^7$ directly, without calculating the whole expansion.
Calculate the coefficient of the term containing $x^3$ in the expansion of the binomial $(x - 4)^5$.
Calculate the coefficient of the term containing $x^3$ in the expansion of the binomial $(x - 4)^5$.
Fractions, Radicals and Negative Exponents
The general term works the same when a piece of the binomial is a fraction or a radical. First rewrite each piece with exponents, then add exponents when you multiply and multiply exponents when you raise a power to a power.
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 $\left(x^2 - \frac{1}{x}\right)^9$.
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 constant term with respect to $x$ in the expansion of the binomial $\left(\sqrt[3]{x^2} - \frac{1}{\sqrt{x}}\right)^{14}$ and determine what position this term…
Find the constant term with respect to $x$ in the expansion of the binomial $\left(\sqrt[3]{x^2} - \frac{1}{\sqrt{x}}\right)^{14}$ and determine what position this term occupies in the expansion ordered by decreasing powers of the first term.
Coefficients of a Given Power and Products
When a question asks for the coefficient of a particular power of $x$, follow the same four steps every time.
To find a coefficient in a product of two expansions, expand each one (or its needed terms), then add up every pair of terms whose exponents sum to the target.
Find the numerical coefficient of $x^4$ in the expansion of the binomial $\left(2x + \frac{1}{x^2}\right)^7$.
Find the numerical coefficient of $x^4$ in the expansion of the binomial $\left(2x + \frac{1}{x^2}\right)^7$.
Calculate the coefficient of $x^5$ in the complete expansion of the product of polynomials $(1 + x)^4 \cdot (1 - 2x)^5$.
Calculate the coefficient of $x^5$ in the complete expansion of the product of polynomials $(1 + x)^4 \cdot (1 - 2x)^5$.
Finding Unknown Parameters
Sometimes the binomial has an unknown number in it, and a given term tells you its value. Use the general term to write an equation, then solve.
Find the value of the real parameter $k$ so that the coefficient of the term in $x^3$ in the expansion of the binomial $\left(x^2 + \frac{k}{x}\right)^6$ is exactly…
Find the value of the real parameter $k$ so that the coefficient of the term in $x^3$ in the expansion of the binomial $\left(x^2 + \frac{k}{x}\right)^6$ is exactly equal to $-160$.
Determine the positive real values of the parameters $a$ and $b$ in the binomial $(ax + b)^5$ knowing that the second term of its expansion is $240x^4$ and the third…
Determine the positive real values of the parameters $a$ and $b$ in the binomial $(ax + b)^5$ knowing that the second term of its expansion is $240x^4$ and the third term of its expansion is $720x^3$.
Properties of Binomial Coefficients
Three properties save a lot of work.
Counting meaning: a set of $n$ items has $\binom{n}{k}$ subsets of size $k$, so $2^n$ subsets in all (including the empty set and the whole set). A pizza place with 6 toppings lets you build $2^6 = 64$ different pizzas, counting plain cheese.
Prove that the sum of all the binomial coefficients in the same row $n$ satisfies $\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} = 2^n$ using the binomial expansion…
Prove that the sum of all the binomial coefficients in the same row $n$ satisfies $\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} = 2^n$ using the binomial expansion 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.
Prove the fundamental property of binomial coefficients $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$ using the algebraic definition with factorials. Subsequently,…
Prove the fundamental property of binomial coefficients $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$ using the algebraic definition with factorials. Subsequently, use the properties of binomial coefficients to solve the following system of equations for natural numbers $x$ and $y$:
$\binom{x}{y} = \binom{x}{y+1}$
$4x - 5y = 19$
Equations with Binomial Coefficients
To solve an equation with $\binom{x}{k}$ or factorials, rewrite everything as polynomials in $x$, solve, then keep only solutions that make sense: $x$ must be a whole number with $x \ge k$.
Find the value of the natural number $x$ that satisfies the following equation with binomial coefficients: $\binom{x}{2} = 21$.
Find the value of the natural number $x$ that satisfies the following equation with binomial coefficients: $\binom{x}{2} = 21$.
Solve the following equation involving binomial coefficients, finding the value of the natural number $x$: $\binom{x}{3} + \binom{x}{2} = 4 \binom{x}{1}$.
Solve the following equation involving binomial coefficients, finding the value of the natural number $x$: $\binom{x}{3} + \binom{x}{2} = 4 \binom{x}{1}$.
Solve the following algebraic equation involving binomial coefficients and factorials, determining the value of the natural number $x$:…
Solve the following algebraic equation involving binomial coefficients and factorials, determining the value of the natural number $x$: $\frac{(x+1)!}{(x-1)!} - 2\binom{x}{2} = 14 + \binom{x-1}{1}$.
Binomial Probability
Suppose each trial has two outcomes: success with probability $p$ and failure with probability $q = 1 - p$. In $n$ independent trials, the probability of exactly $k$ successes is
$$\binom{n}{k}\,p^k\,q^{\,n-k}$$This is the general term of $(p+q)^n$. The $\binom{n}{k}$ counts which $k$ of the $n$ trials are the successes. Because $p + q = 1$, all the terms together add to $(p+q)^n = 1^n = 1$.
In a population genetics study, the theoretical distribution of certain genotypes in a generation is given by the expansion of the binomial $(p + q)^4$, where $p$ and…
In a population genetics study, the theoretical distribution of certain genotypes in a generation is given by the expansion of the binomial $(p + q)^4$, where $p$ and $q$ are the allele frequencies. If $p = 0,6$ and $q = 0,4$, calculate the numerical value of the term containing the expression $p^2q^2$ and interpret its mathematical meaning within the expansion.
In the quality control of an electronic components factory, the probability that a batch of $n$ components has exactly $k$ defective ones is given by the general term…
In the quality control of an electronic components factory, the probability that a batch of $n$ components has exactly $k$ defective ones is given by the general term $\binom{n}{k} p^k (1-p)^{n-k}$, where $p$ is the probability that an individual component is defective. If in a batch of $10$ components, the probability of finding exactly $4$ defective ones is twice that of finding $3$ defective ones, set up the corresponding equation and find the exact value of $p$.
Approximations and Models
When the second piece of a binomial is tiny, its higher powers are much tinier. Keep only the first few terms of $(1 + \varepsilon)^n$:
$$(1+\varepsilon)^n = 1 + n\varepsilon + \binom{n}{2}\varepsilon^2 + \binom{n}{3}\varepsilon^3 + \dots \approx 1 + n\varepsilon$$The curve $y = (1+x)^5$ hugs its approximations near $x = 0$, but drifts away as $x$ gets larger. Notice how the quadratic approximation stays closer than the linear one.
This is the same pattern you need for expansions like $(1 + \cos\theta)^n$: treat $\cos\theta$ as the variable, expand, and only then substitute a value. Likewise, a cube whose edge grows from $L$ to $L(1+\varepsilon)$ has new volume $L^3(1+\varepsilon)^3 = L^3(1 + 3\varepsilon + 3\varepsilon^2 + \varepsilon^3)$. If $\varepsilon$ is tiny, the $\varepsilon^2$ and $\varepsilon^3$ terms are negligible.
In economics, the final capital accumulated after $5$ years at an annual compound interest $r$ is calculated using the formula $C = C_0(1 + r)^5$. Use the binomial…
In economics, the final capital accumulated after $5$ years at an annual compound interest $r$ 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.
The volume of a metal cube with edge $L$ at a certain temperature is $V = L^3$. When heated, its edge expands linearly to measure $L(1 + \alpha \Delta T)$. Expand the…
The volume of a metal cube with edge $L$ at a certain temperature is $V = L^3$. When heated, its edge expands linearly to measure $L(1 + \alpha \Delta T)$. Expand the expression for the new volume using the binomial theorem and, knowing that the product $\alpha \Delta T$ is a very small value (on the order of $10^{-5}$), justify mathematically why the increase in volume $\Delta V$ can be approximated by the expression $3 L^3 \alpha \Delta T$.
In an optics experiment, the light intensity in a multiple interference pattern requires calculating the expansion of the binomial $(1 + \cos \theta)^n$. For a system of…
In an optics experiment, the light intensity in a multiple interference pattern requires calculating the expansion of the binomial $(1 + \cos \theta)^n$. For a system of $6$ slits ($n=6$), expand the expression indicating the binomial coefficients and determine the coefficient of the term containing $\cos^4 \theta$. If at a point on the screen it is true that $\cos \theta = \frac{1}{2}$, what is the exact value of the sum of the last three terms of this expansion?
Back to the start: Emma's probability of exactly $3$ makes in $5$ shots is $\binom{5}{3}(0.7)^3(0.3)^2 = 0.3087$. It's one term of $(0.7 + 0.3)^5 = 1$, found without expanding anything else.