Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ C(7, r) $$

Step-by-Step Solution

Answer

The general formula for combinations with 7 elements is: \(C(7, r) = \frac{7!}{r!(7-r)!}\)

Step 1: Identify the values of n and r

We have \(n=7\), and we need to find \(C(7,r)\) for a given value of \(r\).

Step 2: Recall the combination formula

We know that the combination formula is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Step 3: Substitute the values of n and r into the formula

To find \(C(7,r)\) for any given value of \(r\), we need to substitute the values of \(n=7\) and \(r\) into the combination formula: \[ C(7, r) = \frac{7!}{r!(7-r)!} \] At this point, we can't simplify any further without knowing the value of \(r\). If the student is asked to find \(C(7,r)\) for a specific value of \(r\), they can now substitute the given value for \(r\) into the formula and calculate the result. If the question wants the general formula for combinations with 7 elements, then the result is: \[ C(7, r) = \frac{7!}{r!(7-r)!} \]

Key Concepts

Combination Formula

The concept of combinations is an important topic within combinatorics, which is a branch of mathematics dealing with the counting, arrangement, and combination of objects. When we talk about combinations, we refer to the selection of items from a larger set, where the order of the selection does not matter. For example, selecting a team of 3 members out of 10 people without caring about the lineup.

The formula to calculate the number of combinations is represented as:

• \(C(n, r) = \frac{n!}{r!(n-r)!}\)

Here:

• \(n\) is the total number of items to choose from.
• \(r\) is the number of items to choose.
• \(!\) signifies factorial, which is key to understanding this formula.

So, using this formula will help you calculate different combinations possible for choosing \(r\) objects from \(n\) originally available objects. It is crucial to recognize that combinations differ from permutations, where the order does matter.

Factorials

Factorials play a vital role in calculating combinations, and they are represented with an exclamation mark (!). The factorial of a number \(n\), denoted by \(n!\), is the product of all positive integers up to \(n\).

For example:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(1! = 1\)
• \(0! = 1\) by definition, even though it might seem unusual.

Factorials grow rapidly, meaning that the numbers get very large as \(n\) increases. This is why they are so useful in combinations, allowing precise calculations of possible selections without having to list and count each possibility manually. Understanding how to manipulate factorials simplifies much of what you do in combinatorial calculations.

Binomial Coefficients

Binomial coefficients are another essential component in combinatorics, represented by the formula for combinations \(C(n, r)\). They appear prominently in the binomial theorem, which describes the algebraic expansion of powers of a binomial (like \((a + b)^n\)).

In the context of combinations, the binomial coefficient \(C(n, r)\) represents the number of ways to choose \(r\) items from a set of \(n\) items, and it is written within a pair of vertical lines like this: \(\binom{n}{r}\).

• Example: The expression \(\binom{7}{r}\) is equivalent to the combination formula \(C(7, r)\).

These coefficients appear as the constants in the expansion of a binomial raised to any power and have wide-ranging applications beyond simple counting problems, such as probability theory and algorithm designs. Remembering their notations and property of symmetry, \(\binom{n}{r} = \binom{n}{n-r}\), can make solving problems involving combinatorial identities or expansions much easier.