Problem 18 - Algebra and Trigonometry

Question

\(13-20=\) Evaluate the expression. $$ \left(\begin{array}{l}{5} \\ {2}\end{array}\right)\left(\begin{array}{l}{5} \\\ {3}\end{array}\right) $$

Step-by-Step Solution

Answer

The evaluated expression is 100.

Step 1: Identify the Binomial Coefficient

Recognize that the expression involves two binomial coefficients: \( \binom{5}{2} \) and \( \binom{5}{3} \). A binomial coefficient is defined as \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Step 2: Calculate \( \binom{5}{2} \)

Substitute \( n = 5 \) and \( k = 2 \) into the binomial coefficient formula: \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10 \].

Step 3: Calculate \( \binom{5}{3} \)

Substitute \( n = 5 \) and \( k = 3 \) into the binomial coefficient formula: \[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4 \times 3!}{3 \times 2 \times 1 \cdot 2!} = \frac{20}{6} = 10 \].

Step 4: Multiply the Binomial Coefficients

Multiply the results of the two binomial coefficients: \[ 10 \times 10 = 100 \].

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects. It's like solving puzzles where we figure out the number of ways to place or choose objects without considering their order.

In this exercise, we use the concept of binomial coefficients, which are a part of combinatorics, to evaluate mathematical expressions. A binomial coefficient, expressed as \( \binom{n}{k} \), describes the number of ways to choose \( k \) objects from a collection of \( n \) objects without regard to the order.

There are some key points to remember about combinatorics and binomial coefficients:

• If order doesn't matter, it's a combination.
• If order matters, it's a permutation (although not needed in this exercise).
• Binomial coefficients are useful for expressions like \( (x+y)^n \), known as binomial expansions.

This exercise simplifies combinatorics to use just the binomial coefficients, allowing us to multiply their values and solve the equation.

Factorial notation

Factorial notation is a mathematical way to express the product of an integer and all the positive integers below it. It's denoted by the symbol \( ! \) (exclamation point).

For instance, the factorial of 5, written as \( 5! \), represents \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are the building blocks of the binomial coefficient formula, which involves dividing factorials to count combinations efficiently.

In our exercise, the binomial coefficient formula is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Here, we calculate how many different ways we can select \( k \) items from \( n \) items, using factorials to simplify the arithmetic.

Remember:

• Factorials grow very fast. For example, \( 3! = 6 \), but \( 10! = 3,628,800 \).
• The factorial of zero is \( 0! = 1 \), a unique case important for calculations.
• Factorials are used widely in permutations and combinations to help us find totals accurately.

Mathematical expressions

Mathematical expressions are combinations of numbers, symbols, and operators that represent a value or relationship. Expressions can involve addition, subtraction, multiplication, division, and other operations.

In the given exercise, the mathematical expression consists of multiplying two binomial coefficients: \( \binom{5}{2} \) and \( \binom{5}{3} \). Solving this involves substituting values into the binomial coefficient formula, performing calculations, and then multiplying the results together.

Breaking down expressions into simpler parts makes them easier to understand:

• Identify each part of the expression (like terms) and work on them separately.
• Use mechanisms like parentheses to signal which operations to perform first.
• Combine or simplify parts step by step, ensuring each calculation is accurate.

Solving expressions involves recognizing components, simplifying calculations, and verifying that the solution is consistent with the rules of arithmetic and algebra.