Newton's Binomial Theorem

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

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

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

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.

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 $(x - 4)^5$.

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.

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 $720x^3$.

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

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.

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 $20\,000$ 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 $3 L^3 \alpha \Delta T$.

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 $p$.