Problem 18 - Precalculus with Limits

Question

In Exercises 15 - 18, evaluate using Pascals Triangle. \( _10C_2 \)

Step-by-Step Solution

Answer

The value of \( _10C_2 \) is 45 according to Pascal's Triangle.

Step 1: Understand the combination notation

The combination notation \( _nC_k \) denotes the number of ways to choose 'k' items out of 'n'. Here, this is the combination \( _10C_2 \), which means 2 items are chosen from 10.

Step 2: Locate the row from Pascal's Triangle

Pascal's Triangle starts with row 0 at the top, and each row represents the coefficients of the expanded powers of a binomial. Locate the 10th row, considering that the counting starts from 0.

Step 3: Identify the element

In the identified row, count from left to right to locate the third element (considering that counting starts from 0). This element provides the value for \( _10C_2 \) in Pascal's Triangle.

Key Concepts

Combination Notation

Combination notation is a mathematical way to express the number of possible groupings of items. It's often denoted as \( _nC_k \) or \( C(n, k) \). In this notation, \( n \) stands for the total number of items, and \( k \) is the number of items to choose. The order in which you choose the items does not matter, so it differs from permutations where order does matter.
For example, \( _{10}C_2 \) means we want to know how many ways there are to choose 2 items from a set of 10 items.
This concept is crucial in statistics, probability, and various mathematical computations because it helps calculate outcomes where the sequence or arrangement isn't essential.

• Choose the number of ways for a set size (\( n \))
• Decide how many to select (\( k \))
• Calculate without regard to order

Binomial Coefficients

Binomial coefficients are an essential concept in algebra and combinatorics. They appear in the binomial theorem, which expands expressions such as \( (x + y)^n \). Each binomial coefficient corresponds to a combination and is represented using combination notation.
The binomial coefficient \( _nC_k \) is calculated using the formula \[ _nC_k = \frac{n!}{k!(n-k)!} \]
where \(!\) denotes a factorial, which is the product of an integer and all the integers below it down to one.
Let's break it down using \( _{10}C_2 \):

• Calculate the factorials: \( 10! = 10 \times 9 \times 8 \times ... \times 1 \)
• Compute \( 2! = 2 \times 1 \)
• Compute \( (10-2)! = 8! \)
• Substitute these into the formula to find \(_{10}C_2 = \frac{10!}{2! \times 8!} = \frac{10 \times 9}{2 \times 1} = 45\)

This result is how many different ways you can choose 2 items from 10, aligning with its binomial coefficient.

Combinatorics

Combinatorics is a branch of mathematics that studies counting, arrangements, and combinations of objects. It is especially useful for problems involving Pascal's Triangle, permutations, and combinations.
When dealing with Pascal's Triangle, each row relates to a set of binomial coefficients. These coefficients provide quick answers to problems of choosing \( k \) items from \( n \) without explicitly calculating factorials, like in our \( _{10}C_2 \) example.

• Understand the underlying structure of counting problems
• Apply Pascal's Triangle for efficient calculation
• Explore arrangements that do not regard order as significant

Combinatorics is invaluable in deciding probabilities, optimizing designs, and solving logical puzzles. By mastering these concepts, you can solve real-world problems ranging from simple selection tasks to complex network designs.