Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ C(6,6) $$

Step-by-Step Solution

Answer

The given expression is C(6,6), which represents the number of ways to choose all 6 items from a set of 6. Using the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), we get \(C(6, 6) = \frac{6!}{6!(6-6)!} = \frac{720}{720 \times 1} = 1\). Therefore, there is only 1 way to choose all 6 items from a set of 6 items.

Step 1: Identify n and k

We have the expression C(6, 6), which means n = 6 (the total number of items) and k = 6 (the number of items to choose).

Step 2: Evaluate factorials

Now, we need to find the factorials for n, k, and (n-k). In our case, n = 6 and k = 6, so (n-k) = 6 - 6 = 0. 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 0! = 1 (by definition, the factorial of 0 is 1)

Step 3: Calculate C(n, k)

Now, we can substitute the values of n!, k!, and (n-k)! into the combination formula: \(C(n, k) = \frac{n!}{k!(n-k)!}\) C(6, 6) = \(\frac{720}{720 × 1}\)

Step 4: Simplify the expression

Now that we have the values of the factorials, we can simplify the expression: C(6, 6) = \(\frac{720}{720 × 1}\) = \(\frac{720}{720}\) = 1 So, the expression C(6, 6) equals 1, meaning there is only one way to choose all 6 items from a set of 6 items.

Key Concepts

Factorials

Factorials are essential building blocks in combinatorics. They are used to determine the number of ways to arrange a set of items. To calculate a factorial for a number, you multiply it by every positive integer below it, down to 1. For instance, if you need the factorial of 6, it would be 6 × 5 × 4 × 3 × 2 × 1, which equals 720. Factorials express the total number of permutations possible in a set of numbers.
Factorials are particularly instrumental when dealing with permutations and combinations, as they simplify complex problems by using a relatively straightforward calculation. One special case is the factorial of zero, denoted as 0! In combinatorial calculations, the value of 0! is always defined as 1. This unique property is based on the idea of an empty arrangement, which by convention, is a single complete arrangement.

Combination formula

In combinatorics, the combination formula helps you determine how many ways you can choose a subset of items from a larger set, without regard to the order of selection. This is different from permutations, where the order matters. The combination formula is represented as:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
In this formula, \(n\) is the total number of items, \(k\) is the number of items to choose, and \(!\) denotes the factorial operation. The numerator \(n!\) is the factorial of the total number of items, while the denominator consists of the factorials of \(k\) and \(n-k\).
When calculating \(C(6, 6)\), since you are selecting all items, the formula devolves into \(\frac{6!}{6! \times 0!}\). Simplifying gives \(1\), emphasizing that there is exactly one way to choose all six items from a set of six items.

Permutations

Permutations play a pivotal role in combinatorial mathematics. They refer to the different ways you can arrange items where the order of arrangement matters. For example, imagine trying to determine how many ways you can arrange a set of books on a shelf. Each arrangement signifies a unique permutation.
Permutations are calculated using factorials. So, for a set of \(n\) items, the number of permutations is given by \(n!\). In situations where you're only choosing a subset \(k\) from \(n\) items, the number of permutations of \(k\) items is calculated using:

• \[ P(n, k) = \frac{n!}{(n-k)!} \]

Unlike combinations, permutations do care about the sequence, making them useful for problems like seating arrangements, password generation, and task scheduling. Understanding permutations helps grasp the intricate ways items can be sequentially ordered to satisfy certain criteria.