Problem 83 - Algebra 2

Question

Evaluate each expression. \(_{6} \mathrm{C}_{5}\)

Step-by-Step Solution

Answer

The result of the given combination problem, _6C5, is 6.

Step 1: Find Values

In our given expression _6C5, n represents the total number of items which is 6 and r represents the number of items to select, which is 5.

Step 2: Factorial Calculation

Before we substitute the values into the combination formula, let's calculate the factorials: 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720, 5! = 5 * 4 * 3 * 2 * 1 = 120 and 1! which is simply 1.

Step 3: Substitute Values into Formula

Place the calculated factorials into the combination formula: _6C5 = \(\frac{6!}{((6-5)!)(5!)}\). After substitution, we get: _6C5 = \(\frac{720}{1 * 120}\).

Step 4: Calculate the Result

Performing the calculation, we finally get: _6C5 = 6.

Key Concepts

Factorials

Factorials are fundamental in the study of permutations and combinations. They are used to express the product of a sequence of descending natural numbers. The factorial of a number is denoted by the symbol '!', so for a positive integer n, the factorial is defined as \[ n! = n \times (n-1) \times (n-2) \times \, ... \, \times 2 \times 1 \]Factorials dramatically increase as the number grows because you multiply the number by every positive integer below it.

• For example, the factorial of 3 (written as 3!) is: \[ 3! = 3 \times 2 \times 1 = 6 \]
• Similarly, 5! is 120 because: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

These numbers are central to calculations in permutations and combinations where the arrangement or combination of items is considered. Even small factorials play a huge role in simplifying complex expressions and solving problems involving large datasets or sequences.

Permutation and Combination

Permutation and combination are two different ways to count arrangements of a set of items. Permutations focus on the arrangement of items where the order matters, while combinations focus on the selection of items where order does not matter.

• Permutation: When you need to find out how many ways you can arrange a set of items, use permutations. For instance, arranging 3 books on a shelf considers permutations. \[ nPr = \frac{n!}{(n-r)!} \]
• Combination: When the order of selection doesn’t matter, use combinations. The formula for combinations is: \[ nCr = \frac{n!}{r!(n-r)!} \]

The distinction between permutations and combinations is crucial.In our example, \[ _6C5 \] represents a combination of 6 items taken 5 at a time, meaning how many different groups of 5 can be made from 6 items, without regard for order. The answer, calculated by the formula, highlights the difference in application between permutations and combinations.

Mathematical Expressions

Mathematical expressions are statements combined using numbers, variables, operation symbols, and groups of those symbols representing a specific value.Expressions are essential tools in conveying mathematical ideas and are central to mathematical analysis.Mathematical expressions are often solved using a series of steps:

• Identify known quantities (e.g., numbers or defined variables)
• Apply arithmetic operations as per mathematical rules (e.g., order of operations/BODMAS)
• Use well-known formulas (e.g., combination formula, factorial formula)

For instance, in the expression \[ _6C5 = \frac{6!}{(6-5)! \times 5!} \]substitute known values to simplify the expression. Having calculated factorials as in the solution above, the substitution becomes straightforward.Mathematical expressions can appear complex but often break down into simpler parts using structured problem-solving, which is beneficial for tackling more intricate mathematical challenges.