Problem 80 - Algebra 2

Question

Simplify each expression. $$ _{5} \mathrm{C}_{2}+_{5} \mathrm{C}_{3} $$

Step-by-Step Solution

Answer

The simplified expression of the problem 5C2 + 5C3 is equal to 20.

Step 1: Find the Combination of 5C2

For this step, apply the combination formula, which is n! / (k!*(n-k)!). Hence, 5C2 would be calculated as 5! / (2!(5-2)!) = 5! / (2!*3!) = (5*4*3*2*1) / (2*1 * 3*2*1) = 10.

Step 2: Find the Combination of 5C3

Use the same formula for this step as well. Hence, 5C3 would be calculated as 5! / (3!(5-3)!) = 5! / (3!*2!) = (5*4*3*2*1) / (3*2*1 * 2*1) = 10.

Step 3: Add the results

Now that both combinations are calculated, add them to get the final result. So, 5C2 + 5C3 = 10 + 10 = 20

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that expands expressions raised to a power. When dealing with an expression of the form \((a + b)^n\), it can be expanded into a series involving terms in the form of combinations. The theorem states that:

• Each term in the expansion is a combination.
• The coefficients of these terms are determined by binomial coefficients, denoted as \(_{n}\mathrm{C}_{k}\).

These coefficients are very crucial in calculating the individual terms of the expansion. They tell you how to distribute the powers of \(a\) and \(b\) across different terms.
The sums of combinations like \(_{5} \mathrm{C}_{2} + _{5} \mathrm{C}_{3}\) in a simplified format is a key component to understand the expansions of binomials.

Factorial Notation

Factorial notation is an essential mathematical concept used to simplify expressions involving permutations, combinations, and more. It is represented by \(n!\), and it means the product of an integer and all the integers below it. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

• Factorial notation is commonly used in the denominator of fractions when calculating combinations.
• It helps in reducing lengthy calculations into more manageable terms.

In our given example, both \(_{5} \mathrm{C}_{2} \) and \(_{5} \mathrm{C}_{3}\) use factorials like \(2!\) and \(3!\) to simplify calculations. Understanding this notation is key when you're solving problems involving combinations and permutations.

Permutations

Permutations are all about arranging a set of objects in a specific order. Unlike combinations, the order matters in permutations. They calculate the number of ways to arrange \(n\) items in \(k\) ways.
However, in the context of our problem, permutations are not directly used since combinations do not consider order as important. Yet, knowing permutations can give you a deeper understanding of how to distinguish between arrangements where order matters and where it doesn't.

• Factorial notation is also used in permutations to compute the total number of arrangements.
• Understanding permutations can provide insights into when to apply each concept appropriately in problems, especially when differentiating them from combinations.

Basic Arithmetic Operations

Arithmetic operations form the foundation for mathematical calculations, including addition, subtraction, multiplication, and division. In problems relating to combinations:

• Multiplication and division come in when computing factorials and applying the combination formula.
• Addition plays a role when combining results of different operations, as shown by the final step in the original solution where we add \(_{5} \mathrm{C}_{2}\) and \(_{5} \mathrm{C}_{3} \) to get \(20\).

Adding these results demonstrates how basic arithmetic is not just important, but integral to bringing together various components of a problem to find a solution. Having a strong grasp of these operations is essential to excel in more complex mathematical problems.