Problem 10 - College Algebra

Question

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{10} C_{6} $$

Step-by-Step Solution

Answer

The value of \(_{10}C_{6}\) is 210.

Step 1: Identify 'n' and 'r'

In the given expression \(_{10}C_{6}\), our \(n\) value is 10 and the \(r\) value is 6. These are the values we plug into our formula.

Step 2: Calculate n!

First, calculate the factorial of \(n = 10\), represented as \(10!\). This is found by multiplying all whole numbers from 1 to 10, so \(10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800\).

Step 3: Calculate r!

Next, calculate \(r!\) which is \(6!\). This is done by multiplying all numbers from 1 to 6, so \(6! = 6 × 5 × 4 × 3 × 2 × 1 = 720\).

Step 4: Calculate (n-r)!

Now, calculate \((n - r)!\) which is \((10 - 6)!\) or \(4!\). This is done by multiplying all numbers from 1 to 4, so \(4! = 4 × 3 × 2 × 1 = 24\).

Step 5: Apply the combination formula

Now apply the combination formula \(C(n, r) = n! / [r!(n - r)!]\). So, substituting in our calculated factorials, we get \(C(10, 6) = 3,628,800 / (720 × 24)\).

Step 6: Simplify the equation

So now we simplify our equation by doing the arithmetic. \(C(10, 6) = 3,628,800 / 17,280\). Dividing 3,628,800 by 17,280, we get 210. So, \(_{10}C_{6} = 210\).

Key Concepts

Factorial

Factorials are a fundamental concept in combinatorics, which is the branch of mathematics dealing with combinations and arrangements. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number. For example, the factorial of a number \( n \), written as \( n! \), is calculated as:

• \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \)

It's important to note that the factorial of 0 is defined as 1, i.e., \( 0! = 1 \). Factorials grow very quickly as the numbers increase, which is why they are particularly useful in counting and arranging items.

For example, if you wanted to find \( 5! \), you would calculate it as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Similarly, \( 10! \) builds upon this sequence of multiplications. Factorials are key in many mathematical formulas, including those in statistics, probability, and calculus, making them essential to understand when working with permutations and combinations.

Combination Formula

The combination formula is a powerful tool used to determine the number of ways to choose \( r \) items from a set of \( n \) items without regard to the order of selection. This is different from permutations, which consider the order. The formula for a combination is often written as:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

Here, \( n! \) stands for the factorial of \( n \), \( r! \) is the factorial of \( r \), and \((n-r)!\) is the factorial of \( (n-r) \). This formula essentially accounts for removing repetition and ignoring the order of selection.

By using this formula, you can calculate the number of combinations without having to list them all, which is very useful when dealing with large numbers. For instance, choosing 6 items from a set of 10 items can be determined without manually listing or calculating each possible selection. The combination formula simplifies these calculations, saving a lot of time and effort.

nCr

\( nCr \) is a notation that specifically refers to the number of combinations possible when choosing \( r \) elements from a set of \( n \) elements. It is synonymous with the combination formula \( C(n, r) \) and is often used interchangeably.

• The notation \( _{n}C_{r} \) is typically read as "\( n \) choose \( r \)."
• It represents the combinatorial operation used to calculate how many ways you can select \( r \) items from \( n \) without regard to order.

The role of \( nCr \) in combinatorics and statistics cannot be overstated. This simple notation unlocks the ability to solve complex problems involving selection without order. For instance, determining \( _{10}C_{6} \) is much more straightforward when you have a grasp of \( nCr \).

When you encounter problems requiring you to "choose" or "select" without worrying about order, \( nCr \) is the go-to tool. With understanding and practice, you can master its application in various mathematical scenarios.