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

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 $x$: $\binom{x}{3} + \binom{x}{2} = 4 \binom{x}{1}$.

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

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

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 $y$:

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

$4x - 5y = 19$

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.

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.

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.

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.

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 $\left(x^2 - \frac{1}{x}\right)^9$.

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 $k$.

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

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.

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

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.

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 $1000$ € using only the first three terms of the expansion.

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 $p$.