Question

Find the value of \(\mathrm{C}(\mathrm{n}, 0)\).

Short answer

The value of \(\mathrm{C}(\mathrm{n}, 0)\) is always 1 for any given n.

Step-by-step solution

Recall the combination formula

The formula for combinations (also called "n choose k") is given by: \[ C(n, k) = \frac{n!}{k!(n-k)!} \]

Plug in k = 0

In this problem, we are asked to find \(\mathrm{C}(\mathrm{n}, 0),\) so we just need to substitute k with 0 in the above formula: \[ C(n, 0) = \frac{n!}{0!(n-0)!} \]

Simplify the expression

Note that 0! is equal to 1. Therefore, we have: \[ C(n, 0) = \frac{n!}{1 \cdot n!} \] Since the numerator and denominator both contain an n!, we can simply cancel them out: \[ C(n, 0) = 1 \] For any n, choosing zero items out of n will always result in 1 combination (the empty set). So, the value of \(\mathrm{C}(\mathrm{n}, 0)\) is always 1.

Key concepts

Understanding Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics used to find the number of ways to choose a subset of items from a larger set without considering the order. It's often represented as \( C(n, k) \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.
In mathematical terms, the binomial coefficient is expressed through the formula:

\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

This formula helps us calculate how many different combinations are possible when selecting \( k \) items from \( n \) items.
Some key points to remember about binomial coefficients are:

• It's applicable in various fields including probability theory, algebra, and statistics.
• It provides insight into the expansion of binomials by distributing terms.

Using the binomial coefficient, you can explore patterns and make predictions in data arrangements.

The Basics of n Choose k

"n choose k" is a common term used to describe how combinations work. It essentially means selecting \( k \) elements from a set of \( n \) elements.
It is denoted mathematically as \( {n \choose k} \), and it has the same value as the binomial coefficient \( C(n, k) \).

Key insights about "n choose k" include:

• It's not concerned with the order in which the items are selected. This differentiates it from permutations, where order matters.
• The value of \( {n \choose 0} \) is always 1. This represents the empty set or choosing nothing from the set.
• "n choose k" is symmetrical, meaning \( {n \choose k} = {n \choose n-k} \). Therefore, selecting \( k \) items is equivalent to leaving out \( n-k \) items.

This concept underpins many mathematical problems and helps simplify thinking about arrangements and selections.

Exploring Factorial

The factorial, denoted by \(!\), is a product of all positive integers up to a specified number. For example, the factorial of 5, written as \(5!\), is calculated as:

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Factorials are integral to the calculation of combinations and permutations because they simplify division by arranging terms.

Some key characteristics of factorials are:

• By definition, \(0! = 1\). This is a crucial point in combination problems, especially when we calculate \( C(n, 0) \).
• Factorials grow very large, very fast. This means they are used often in computational problems where heavy calculations are needed.
• They play a critical role in binomial theorem and series expansions, showcasing patterns in algebraic expressions.

Understanding factorials will enhance your ability to solve complex combinatorial problems efficiently.