Problem 77 - Algebra 2

Question

Simplify each expression. $$ _{7} \mathrm{C}_{3} $$

Step-by-Step Solution

Answer

The simplified expression for \( _{7}\mathrm{C}_{3} \) is 35.

Step 1: Write down the combination formula

First, state the combination formula: \( _{n}\mathrm{C}_{r} = \frac{n!}{r!(n - r)!} \).

Step 2: Substitute the given values into the combination formula

Next, substitute the given values \( n = 7 \) and \( r = 3 \) into the formula: \( _{7}\mathrm{C}_{3} = \frac{7!}{3!(7 - 3)!} = \frac{7!}{3!4!} \).

Step 3: Simplify the factorials in the numerator and the denominator

Simplify the factorials in the numerator and denominator: \( \frac{7!}{3!4!} = \frac{7 * 6 * 5 * 4!}{3! * 4!} \). The factorials \( 4! \) in the numerator and denominator can be cancelled out. This results in: \( \frac{7 * 6 * 5}{3!} = \frac{7 * 6 * 5}{3 * 2 * 1} \).

Step 4: Cancel out terms and compute the final value

Cancel out the 3 in the denominator with the 6 in the numerator (as 6 is a multiple of 3) to get \( \frac{7 * 2 * 5}{2 * 1} \). The 2's can be cancelled out to finally result in 35. Therefore, \( _{7}\mathrm{C}_{3} = 35 \).

Key Concepts

Factorials

In mathematics, a factorial is a function that multiplies a series of descending natural numbers. It's denoted by an exclamation point (!). For any given positive integer \( n \), the factorial is the product of all positive integers less than or equal to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials are commonly used in permutations, combinations, and probability calculations.

It’s essential to understand that \( 0! \) is defined as 1. This might seem counterintuitive, but it ensures that calculations involving factorials remain consistent. Factorials grow rapidly—a simple factor of \( n! \) means multiplying \( n \) by all smaller positive integers.
• **Usage in Combinations**: Factorials are used to find combinations—a way to select items from a larger pool without regard to order. In our exercise, to compute \( _{7}\mathrm{C}_{3} \), you simplify the expression \( \frac{7!}{3! \, 4!} \).
• **Simplifying Expressions**: In simplifying these expressions, recognizing common factorials in the numerator and denominator helps in cancelling terms. For example, the \( 4! \) in both parts of the fraction cancels out, simplifying the equation and making calculation easier.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that provides a formula for expanding expressions that are raised to a power. The binomial theorem states that for any positive integer \( n \),\( (x + y)^n = \sum_{k=0}^n \binom{n}{k} \, x^{n-k} \, y^k \), where \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient.

The binomial coefficient, denoted as \( \binom{n}{k} \), is critical here. It's essentially a combination, like \( _{n}\mathrm{C}_{r} \), and represents the number of ways to choose \( k \) elements from \( n \) elements. Our problem of \( _{7}\mathrm{C}_{3} \) is a direct application of this concept.
• **Understanding Binomial Coefficients**: These coefficients appear as the numbers in Pascal's triangle and are used to determine the coefficients in the expansion of binomial expressions.
• **Link to Combinations**: The coefficients in the theorem directly relate to combinations, showing they’re intrinsic to expanding polynomial expressions efficiently without manually multiplying each term.

Probability

Probability is a measure of the likelihood that an event will occur. It ranges between 0 (impossible event) and 1 (certain event). In probability theory, combinations are crucial because they help you count the number of ways an event can occur, which is essential for calculating probabilities.

In situations where order does not matter, combinations are often paired with probability to calculate events. Take, for example, a lottery scenario where you're picking numbers without concern for the sequence.
• **Using Combinations in Probability**: When calculating probabilities, we often use combinations to determine how many outcomes satisfy an event condition. For example, the expression \( _{7}\mathrm{C}_{3} \) might be used to determine how many ways we can select 3 items from 7, with each selection equally likely.
• **Practical Applications**: Whether determining the probability of winning a card game or assessing experimental data, understanding how to use combinations effectively optimizes problem-solving processes.
By mastering how combinations, facilitated by factorials, operate within probability, students gain a robust toolkit for tackling a wide range of mathematical problems.