Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ C(5,0) $$

Step-by-Step Solution

Answer

\( C(5,0) = 1 \)

Step 1: Recall the formula for combinations

The formula for combinations is given by: $$ C(n, k) = \binom{n}{k} = \frac{n!}{k! (n - k)!} $$ Here, n! (n factorial) represents the product of all positive integers up to n, and C(n, k) is the number of ways to choose k elements from a set of n elements.

Step 2: Plug in the values

In the given problem, we have n = 5 and k = 0. Plugging these values into the combination formula, we get: $$ C(5, 0) = \frac{5!}{0! (5 - 0)!} $$

Step 3: Evaluate the factorials

Recall that the factorial of 0 is 1, i.e., 0! = 1. Now, let's calculate the factorials: 5! = 5 × 4 × 3 × 2 × 1 = 120 0! = 1 (5 - 0)! = 5! = 120

Step 4: Substitute the factorials and calculate the result

Substitute the calculated factorial values back into the formula: $$ C(5,0) = \frac{120}{1 \times 120} $$

Step 5: Simplify the expression and find the result

Now, let's simplify the expression: $$ C(5,0) = \frac{120}{120} = 1 $$ Hence, there is only 1 way to choose 0 elements from a set of 5 elements, which means C(5, 0) = 1.

Key Concepts

Factorial Notation

When we step into the world of mathematics, we often encounter an intriguing symbol: '!'. This symbol represents factorial notation, a fundamental concept particularly useful in combinatorics, which is the study of counting. A factorial is the product of an integer and all the positive integers below it. For instance, the factorial of 5, denoted as \(5!\), is calculated as follows:
\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]
The value of \(0!\) might seem unusual at first, as one might question what it means to multiply no numbers together. However, for the sake of mathematical conventions and practical uses in formulas, \(0!\) is defined to be 1. This helps in simplifying expressions and makes the formulas for combinations and permutations work seamlessly, even when choosing zero objects from a set.
Understanding factorial notation is not just about multiplying a series of descending numbers; it involves recognizing its ubiquitous presence in various mathematical contexts, particularly in computing permutations and combinations, which quantify possibilities and choices. As we delve into other concepts of combinatorics, keep in mind how the factorial underpins many formulas you'll encounter.

Binomial Coefficient

A binomial coefficient appears often in mathematical expressions, especially when dealing with combinations and the expansion of binomial expressions. It is represented as \( \binom{n}{k} \), which can be read as 'n choose k'. The binomial coefficient is a way of determining the number of ways to choose a subset of k elements from a larger set of n elements, regardless of the order of selection. Using the factorial notation we previously described, the binomial coefficient is defined by the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
For example, the question of how many different committees of 3 people can be formed from a group of 10 people can be answered using the binomial coefficient \( \binom{10}{3} \). This coefficient, while rooted in the practical context of selection, also appears in the expansion of \((a+b)^n\), known as the binomial theorem, where it determines the coefficients of the terms in the expanded polynomial. The elegant relationship between the binomial coefficient and concepts of choice and possibility is a cornerstone of combinatorial mathematics and probability theory.

Permutations and Combinations

Grasping the distinction and applications of permutations and combinations opens a new dimension in understanding organizational possibilities. Permutations are about arranging objects in a specific order. The number of different permutations of n objects taken k at a time is given by:
\[P(n, k) = \frac{n!}{(n-k)!}\]
This formula accounts for the order in which the objects are arranged.
On the other hand, combinations, as discussed earlier, deal with selecting objects without regard to the order. The mathematical expression for combinations, \(C(n, k)\), simplifies to the binomial coefficient:
\[C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}\]
For instance, if you're interested in how many different ways you can choose 2 fruits from a basket of 5 (assuming each fruit is distinct), the order in which you select the fruit doesn't matter. You're dealing with combinations.
In simpler terms, permutations are for lists (where order matters), and combinations are for groups (where order doesn't matter). Students might find it easier to remember through an everyday analogy: Permutations are like arranging books on a shelf (where the order is important), while combinations are like putting together a fruit salad (where the order of fruit doesn't matter). Learning to calculate each correctly is an essential skill for problems involving probability, games, and any situation where context dictates whether the order of selection or arrangement is important.