Exercises: Binomial theorem

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

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.

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

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

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

Simplify the following algebraic expression with factorials completely: $\frac{(n+2)!}{n! \cdot (n+1)}$.

Find the value of the natural number $x$ that satisfies the following equation with binomial coefficients: $\binom{x}{2} = 21$.

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 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.

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 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.

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

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 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 $-160$.

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 $p$.

Calculate the coefficient of $x^5$ in the complete 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 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.

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?

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

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 $720x^3$.