Problem 26 - College Algebra

Question

Evaluate each expression. $$ \frac{_{10} C_{3}}{_6 C_{4}}-\frac{46 !}{44 !} $$

Step-by-Step Solution

Answer

-2069.9984

Step 1: Calculate the Combinations

First, calculate the combinations separately using the formula \(\frac{n!}{r!(n-r)!}\). The first combination \(_{10}C_{3}=\frac{10!}{3!(10-3)!}=\frac{720}{6*5040}=0.024\). The second combination \(_6C_{4}=\frac{6!}{4!(6-4)!}=\frac{720}{24*2}=15\).

Step 2: Evaluate the Combination Expression

Next, compute the expression \(\frac{_{10}C_{3}}{_{6}C_{4}}= \frac{0.024}{15}=0.0016\)

Step 3: Calculate the Factorials

Then, calculate \(46!\) and \(44!\). Remember that \(46!=46*45*44!\). Hence, \(46!/44!=(46*45)=2070\).

Step 4: Evaluate the Whole Expression

Finally, subtract the result of the factorial from that of the combination expression: \(0.0016-2070=-2069.9984\).

Key Concepts

Combinations

Combinations are a fundamental part of combinatorics and are useful when you want to count how many different groups can be formed from a larger set. In mathematics, combinations refer to selecting items from a group, where the order of selection does not matter. To calculate combinations, we use the combination formula: \[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \] Here, \(n\) is the total number of items, and \(r\) represents the number of items to choose. The symbol \(_{n}C_{r}\) denotes the number of combinations.

• For the expression \(_{10}C_{3}\), it means selecting 3 items from 10.
• This uses the formula \(\frac{10!}{3!(10-3)!}\).

This expression is crucial because it helps you evaluate the possible scenarios efficiently without considering each possibility individually, especially when working with larger numbers. Combinatorics often underlies problems in probability and other mathematical fields.

Factorials

Factorials are an essential concept when working with combinations or permutations. A factorial, denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). In the context of our exercise, factorials help compute combinations, as seen in the expression: \[_{6}C_{4} = \frac{6!}{4!(6-4)!}\] Not only are factorials useful in combinatorics, but they also appear in various branches of mathematics, such as calculus and algebra. Another critical moment when we used factorials was simplifying \(\frac{46!}{44!}\).

• Here, notice that \(46!\) is broken down as \(46 \times 45 \times 44!\).
• Since \(44!\) is in both numerator and denominator, it cancels out, simplifying the expression to \(46 \times 45\).

Understanding how to manipulate and compute factorials is key for solving many mathematical problems.

Evaluate Expressions

Evaluating expressions is a process that refers to calculating the value of an algebraic expression by using the rules and operations of mathematics. It's like solving a puzzle by following a series of logical steps. In the given problem, we start with evaluating individual parts, such as combinations and factorials, and then combine these results.

• First, we compute \(_{10}C_{3}\) and \(_{6}C_{4}\) separately.
• Next, we evaluate the combination expression \(\frac{_{10}C_{3}}{_{6}C_{4}}\).
• Then, solve the factorial expression \(\frac{46!}{44!}\).

Finally, complete the complex expression by performing the operations as guided: subtract the evaluated factorial result from the combination expression to get the final solution. The key is to simplify in steps.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the building blocks of more complex equations and play a significant role in various mathematical computations. In our example, the exercise combines several algebraic concepts:

• Expressions like \(\frac{_{10}C_{3}}{_{6}C_{4}}\) involve combinations which are a core part of algebra.
• Factorial expressions \(\frac{46!}{44!}\) further illustrate how multiplication operates in an algebraic context.

The key to working with algebraic expressions is breaking down complex expressions into smaller, manageable parts. By understanding the individual components, you can perform operations like addition, subtraction, multiplication, and division effectively. Learning these skills will help you navigate and solve numerous mathematical problems, spanning from simple algebra to more advanced topics.