Newton's Binomial Theorem

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

Simplify as far as possible the following algebraic expression involving factorials: $\frac{(n+2)!}{n! \times (n+1)}$.

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 value of the natural number $x$ satisfying the following equation involving binomial coefficients: $\binom{x}{2} = 21$.

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

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 $y$:

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

$4x - 5y = 19$

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

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

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 $q = y^2$.

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.

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 $(x - 4)^5$.

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

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 $k$.

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 $-160$.

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 $x$ (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.

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 exponents.

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

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.

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.

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 variables $x$ and $y$ so that the evaluated expression is exactly equal to the sum of its coefficients.

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

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.

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 $3 L^3 \alpha \Delta T$.

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?

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.

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.

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 $p$.

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?