Exercises: Binomial theorem

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

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: $\frac{(n+2)!}{n! \times (n+1)}$.

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.

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 $60\,000$ pesos 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.

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 $(1 + x)^4 \times (1 - 2x)^5$.

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?