Problem 28 - College Algebra

Question

Evaluate each expression. $$ \frac{_{5} C_{1} \cdot_{7} C_{2}}{_{12} C_{3}} $$

Step-by-Step Solution

Answer

The expression is evaluated to be approximately 0.4773

Step 1: Calculate \(_{5} C_{1}\)

The combination formula is given as: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items, and \( k \) is the number of items to choose. So, \(_{5} C_{1} = \frac{5!}{1!(5-1)!} = 5\)

Step 2: Calculate \(_{7} C_{2}\)

By applying the same formula as in Step 1, \(_{7} C_{2} = \frac{7!}{2!(7-2)!} = 21\)

Step 3: Calculate \(_{12} C_{3}\)

Again applying the same formula, \(_{12} C_{3} = \frac{12!}{3!(12-3)!} = 220\)

Step 4: Evaluate the Final Expression

Now, substitute the values from Step 1, Step 2 and Step 3 into the expression and calculate: \[\frac{_{5} C_{1} \cdot_{7} C_{2}}{_{12} C_{3}} = \frac{5 \cdot 21}{220} = \frac{105}{220} = 0.4773 \]

Key Concepts

Factorial Notation

Factorial notation is a helpful mathematical concept when working with combinations, permutations, and various calculations involving sequences. It is denoted by an exclamation mark

• For any positive integer \(n\), the factorial is written as \(n!\), and it means the product of all positive integers less than or equal to \(n\).
• For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

When using the factorial in combination problems, it helps in calculating the number of ways to choose items from a set. Factorials grow very quickly; for instance, while \(3!\) equals 6, \(10!\) escalates to a whopping 3,628,800.
Knowing how to handle factorials efficiently is crucial when working with the combination formula, which is used to find binomial coefficients.

Combination Formula

The combination formula is a mathematical tool used to determine the number of ways a subset can be selected from a larger set, where the order of selection does not matter.
It is often denoted as \(C(n, k)\), \(nCk\), or \(_{n}C_{k}\).

• The formula itself is: \[C(n, k) = \frac{n!}{k!(n-k)!}\]
• Here, \(n\) represents the total number of items in the set, and \(k\) is the number of items to choose.

This formula is built on factorial notation, and it is essential for solving problems like example expressions where we are combining more than one of these calculations.
Using this formula makes it easy to evaluate terms in binomial expansions and other statistical calculations quickly and accurately.

Binomial Coefficient

The binomial coefficient is a central concept in combinatorics and is closely related to the combination formula. It represents the number of ways to choose \(k\) elements from a set of \(n\) elements.

• In simpler terms, it's a way to describe how to pick items from a set without considering the order.
• It is expressed as \(\binom{n}{k}\) or \(C(n, k)\).

The binomial coefficient is a key building block in the binomial theorem, which describes the algebraic expansion of powers of a binomial.
By understanding the binomial coefficient, students can easily solve problems involving choosing subsets from a larger set, as seen in the example exercise. These coefficients also appear in various mathematical contexts, such as probabilities and statistical distributions.