Exercises: Binomial theorem

Calculate the value of the following expression using the properties of factorials and binomial coefficients: $\frac{10!}{8! \cdot 2!} + \binom{7}{3}$.

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.

Write down directly the fourth term in the expansion of the binomial $(x^2 - 2y)^7$, without finding the full expansion.

Use Pascal's triangle (or Tartaglia's triangle) to obtain the coefficients of the sixth row and expand the expression $(m + 2n)^6$.

Find the coefficient of the term containing $x^3$ in the expansion of the binomial $(x - 4)^5$.

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: $\binom{x}{2} = 21$.

Expand the binomial with fractions $\left(x + \frac{1}{2}\right)^3$ and simplify each of the resulting terms.

Sofía is calculating the term in the fourth position ($T_4$) in the expansion of the binomial $(x^2 - 2)^6$. On her exam, she writes the following:

"Since the fourth term is requested, I use $k = 4$:

$T_4 = \binom{6}{4} (x^2)^{6-4} (-2)^4 = 15 \cdot x^4 \cdot 16 = 240x^4$".

1. a) Identify and explain, with reasoning, the conceptual error that Sofía makes when associating the term number with the index $k$ in the general formula $T_{k+1} = \binom{n}{k} a^{n-k} b^k$.
2. b) Correct the solution by calculating step by step the true fourth term of the expansion, clearly stating its sign, its numerical coefficient, and its degree.
3. c) Which term of the expansion did Sofía actually calculate in her procedure?

A telecommunications company in Spain applies an inflation adjustment of $3\,\%$ per year ($0,03$) for $4$ consecutive years to a flat-rate fibre optic plan. The cost after that time ends up multiplied by the factor $(1 + 0,03)^4$.

1. a) Mentally estimate the value of this multiplier factor using only the first two terms of the binomial theorem, $(1 + 0,03)^4 \approx \binom{4}{0} \cdot 1^4 + \binom{4}{1} \cdot 1^3 \cdot 0,03$, and indicate what total percentage increase your estimate represents.
2. b) Expand the power $(1 + 0,03)^4$ completely using the binomial theorem and calculate the exact value of the multiplier factor to four decimal places.
3. c) Check whether the estimate in part a) was an overestimate or an underestimate. Justify why in economics it is usually sufficient to keep only the first two terms when the annual increase is small.

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.

Find the term independent of $x$ (that is, the term that does not contain the variable $x$) in the expansion of the binomial $\left(x^2 - \frac{1}{x}\right)^9$.

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.

Find the numerical coefficient of $x^4$ in the expansion of the binomial $\left(2x + \frac{1}{x^2}\right)^7$.

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.

Solve the following equation involving binomial coefficients, finding the value of the natural number $x$: $\binom{x}{3} + \binom{x}{2} = 4 \binom{x}{1}$.

The volume of a metal cube of edge length $L$ at a certain temperature is $V = L^3$. When heated, its edge expands linearly to a length of $L(1 + \alpha \Delta T)$. Expand the expression for the new volume using the binomial theorem and, knowing that the product $\alpha \Delta T$ is a very small value (of the order of $10^{-5}$), mathematically justify why the volume increase $\Delta V$ can be approximated by the expression $3 L^3 \alpha \Delta T$.

Determine the middle term in the expansion of the binomial $\left(\sqrt{x} - \frac{2}{x}\right)^6$ and simplify the resulting expression.

In a genetics study, the combination of certain alleles is modelled using the mathematical expression $(p + q)^5$. Expand this expression given that the probabilities are $p = 2x$ and $q = y^2$.

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 $x$.

Determine the coefficient of the term containing $x^4$ when expanding the expression $(x - \sqrt{2})^6$. Express the result as an integer.

The intensity of light passing through a series of $4$ photographic filters with a variable attenuation factor $x$ can be approximated by the binomial $(1 - 0,2x)^4$. Expand this expression, arranging the terms in ascending powers.

In the expansion of the binomial $(x + k)^4$, the coefficient of the term in $x^2$ is known to be equal to $54$. Calculate the positive value of the parameter $k$.

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.

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.

A Galton board (or quincunx) is used to demonstrate the binomial distribution. A falling ball has the same probability of going right ($D$) as left ($I$) at each peg. The board in the figure has $4$ rows of pegs, so the ball makes $4$ moves and can land in $5$ bins. Each bin is labeled with the number of moves to the right that the ball has made. The possible paths are modeled by the expansion of $(I + D)^4$.

Look at the figure: the dashed blue line shows a possible path to the bin highlighted in red, which corresponds to $1$ move to the left and $3$ to the right.

1. a) Expand the binomial $(I + D)^4$.

1. b) State algebraically how many different paths lead to the bin highlighted in red.

The National Statistics Institute (INE) conducts an audit on the express parcel service in an autonomous community by analysing a sample of $7$ independent deliveries. Each parcel can arrive on time ($P$) or late ($R$). All possible sample outcomes are modelled algebraically using the expansion of the binomial $(P + R)^7$.

1. a) Write the complete expansion of the binomial using Newton's formula and interpret the meaning in this context of each term of the form $\binom{7}{k} P^{7-k} R^k$.
2. b) How many different sample configurations contain at least $5$ on-time deliveries?
3. c) Regional regulations establish that quality is acceptable if the number of configurations with more on-time deliveries than late deliveries strictly exceeds $50\,\%$ of the total possible cases ($2^7 = 128$). Using the symmetry of the binomial coefficients in Pascal's triangle, determine the exact percentage of favourable configurations and conclude with reasoning whether the regulatory criterion is met.

In the algebraic expansion of the binomial $\left(2x^3 - \frac{k}{x}\right)^8$, where $k$ represents a positive real constant:

1. a) Write the simplified expression for the general term $T_{m+1} = \binom{8}{m} a^{8-m} b^m$ and set up the equation for the exponent of $x$ to determine which position the term independent of $x$ occupies.
2. b) Knowing that the numerical value of this constant term is $7\,168$, calculate with reasoning the exact value of the constant $k$.
3. c) Check algebraically whether there is any term of degree $5$ in the variable $x$ in this expansion, justifying your conclusion from the set of admissible values for the index $m$.

Find the value of the real parameter $k$ such 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 $-160$.

In quality control at an electronic component factory, the probability that a batch of $n$ components has exactly $k$ defective items is given by the general term $\binom{n}{k} p^k (1-p)^{n-k}$, where $p$ is the probability of an individual component being defective. If in a batch of $10$ components, the probability of finding exactly $4$ defective items is twice that of finding $3$ defective items, set up the corresponding equation and find the exact value of $p$.

Calculate the coefficient of $x^5$ in the full expansion of the product of polynomials $(1 + x)^4 \cdot (1 - 2x)^5$.

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 $y$:

$\binom{x}{y} = \binom{x}{y+1}$

$4x - 5y = 19$

Find the term independent of $x$ in the expansion of the binomial $\left(\sqrt[3]{x^2} - \frac{1}{\sqrt{x}}\right)^{14}$ and determine its position in the expansion ordered by descending powers of the first term.

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 showing the binomial coefficients and determine the coefficient of the term containing $\cos^4 \theta$. If at a point on the screen $\cos \theta = \frac{1}{2}$ holds, what is the exact value of the sum of the last three terms of that expansion?

Solve the following algebraic equation involving binomial coefficients and factorials, finding the value of the natural number $x$: $\frac{(x+1)!}{(x-1)!} - 2\binom{x}{2} = 14 + \binom{x-1}{1}$.

Determine the positive real values of the parameters $a$ and $b$ in the binomial $(ax + b)^5$ given that the second term of its expansion is $240x^4$ and the third term of its expansion is $720x^3$.

Find the term independent of $x$ in the expansion of the expression $\left(\frac{3x^2}{2} - \frac{1}{3x}\right)^9$. Show the general formulation of the $k$th term before solving it.

Expand the sum of the binomials $(\sqrt{3} + \sqrt{2})^4 + (\sqrt{3} - \sqrt{2})^4$. Show that, despite containing irrational numbers in its initial formulation, the final result is an integer.

In engineering, the volume tolerance of a cubic container of side $L$ with a manufacturing error $\Delta L$ is given by the expression $(L \pm \Delta L)^3$. Expand this binomial. If $L = 10\ \text{cm}$ and the maximum error is $\Delta L = 0,1\ \text{cm}$, calculate the linear, quadratic, and cubic error terms. What percentage of the total error does the linear term $3L^2\Delta L$ represent?

The goal is to simplify the combinatorial expression $E = \binom{n}{k} + 2\binom{n}{k+1} + \binom{n}{k+2}$, where $n, k \in \mathbb{N}$ with $n \ge k+2$. Two solution methods are proposed:

• Method A (factorial expansion): Express each binomial coefficient according to the formula $\frac{n!}{p!\,(n-p)!}$, find the least common denominator among the factorials, and simplify the resulting fraction.

• Method B (properties of Pascal's triangle): Decompose the middle term as $\binom{n}{k+1} + \binom{n}{k+1}$ and repeatedly apply the identity $\binom{m}{p} + \binom{m}{p+1} = \binom{m+1}{p+1}$.

1. a) Apply Method B, justifying each step in detail to condense the expression $E$ into a single binomial coefficient.
2. b) Explain why Method B is remarkably more efficient and less prone to algebraic errors than Method A in this type of simplification.
3. c) Generalise the reasoning: determine, without resorting to factorials, which single binomial coefficient the sum $\binom{n}{k} + 3\binom{n}{k+1} + 3\binom{n}{k+2} + \binom{n}{k+3}$ is equivalent to, explaining the connection between the coefficients $(1, 3, 3, 1)$ and successive rows of Pascal's triangle.

The public company ADIF manages the high-speed rail lines in Spain. During heatwave episodes recorded by AEMET, certain cubic steel components of the overhead lines, with initial volume $V_0$, experience a relative linear thermal expansion $\varepsilon = \alpha \cdot \Delta T = 4 \cdot 10^{-4}$ (dimensionless). The new volume after expansion is given by $V = V_0(1 + \varepsilon)^3$.

1. a) Expand $(1 + \varepsilon)^3$ using the binomial theorem and express the exact relative volume increase, $\frac{\Delta V}{V_0} = \frac{V - V_0}{V_0}$, in terms of the powers of $\varepsilon$.
2. b) In technical manuals, the linear approximation $\frac{\Delta V}{V_0} \approx 3\varepsilon$ is commonly used. Estimate the bound of the absolute error made with this approximation by summing the omitted terms ($3\varepsilon^2 + \varepsilon^3$), and calculate the percentage that this error represents relative to the approximate value $3\varepsilon$.
3. c) A preliminary technical report claims that "if a more severe heatwave doubles the temperature increase ($\Delta T' = 2\Delta T$), the absolute error of the linear approximation will also double". Prove analytically using the binomial expansion why this statement is false, and determine the factor by which the error incurred is approximately multiplied.