Problem 66 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(\left(\begin{array}{c}100 \\ 2\end{array}\right)\)

Step-by-Step Solution

Answer

The value is 4950.

Step 1: Understanding the Binomial Coefficient

The expression \( \binom{100}{2} \) is a binomial coefficient, which is used to calculate the number of ways to choose 2 items from a set of 100 items, without regard to the order of selection.

Step 2: Apply the Binomial Coefficient Formula

The binomial coefficient is calculated using the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]In this case, \( n = 100 \) and \( k = 2 \).

Step 3: Substitute Values into the Formula

Substitute \( n = 100 \) and \( k = 2 \) into the formula:\[\binom{100}{2} = \frac{100!}{2!(100-2)!}\]This simplifies to:\[\frac{100 \times 99 \times 98!}{2 \times 1 \times 98!}\]

Step 4: Simplify the Expression

Cancel out the \( 98! \) in the numerator and the denominator:\[\frac{100 \times 99}{2 \times 1} = \frac{100 \times 99}{2}\]

Step 5: Calculate the Final Result

Calculate the result by simplifying the remaining fraction:\[\frac{100 \times 99}{2} = \frac{9900}{2} = 4950\]

Key Concepts

Combinatorics

Combinatorics is a fascinating area of mathematics that deals with counting and arranging objects. It helps us find out how many possible ways there are to do something, like arranging books on a shelf or forming a committee. In this context, we're looking at how to choose a specific number of items from a larger set, which is where binomial coefficients come into play.

• Binomial coefficients are a key concept in combinatorics used for understanding combinations.
• They represent the number of ways to choose \( k \) items from \( n \) items without considering the order.
• The expression \( \binom{n}{k} \) is used to denote these combinations.

If you have ever wondered how a committee of 2 people can be chosen from a pool of 100, combinatorics provides the answer. By learning its principles, you gain powerful tools to solve a wide range of practical and theoretical problems.

Factorials

Factorials are instrumental in the calculation of combinations and permutations. The factorial of a number, denoted by \( n! \), is the product of all positive integers up to that number.

• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Factorials build up very quickly, which is both their power and their challenge.
• In binomial coefficients, factorials are used to determine the number of permutations and combinations.

In our exercise, \( 100! \), \( 2! \), and \( 98! \) play a role in calculating how many ways you can choose 2 items from 100, showing how factorials help simplify these large computations. Understanding the swift growth of factorials is crucial for grasping their mathematical impact.

Algebraic Expressions

Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations. In our case, algebra helps simplify the problem, making it easier to solve.

• An algebraic expression like \( \frac{100 \times 99 \times 98!}{2 \times 1 \times 98!} \) looks complex, but algebra allows us to see what can be cancelled or simplified.
• The ability to reduce expressions by cancelling similar terms (like \( 98! \) in both numerator and denominator) demonstrates algebra's power.
• This simplification leads to a manageable expression: \( \frac{100 \times 99}{2} \).

Working with algebraic expressions enables you to focus on the core problem by removing unnecessary complexity. This makes it easier to perform calculations and reach the final result, like finding that there are 4,950 ways to choose 2 items from 100, using clear and logical steps.