Problem 17 - Precalculus with Limits

Question

In Exercises 15 - 18, evaluate using Pascals Triangle. \( _7C_4 \)

Step-by-Step Solution

Answer

The result of \( _7C_4 \) is 35.

Step 1: Identify the structure of Pascal's Triangle

In Pascal's Triangle, each row represents the coefficients of the binomial expansion. The row number (starting from 0) is equivalent to the expansion's power. Each term in the row corresponds to the combination of the row number and term position (also starting from 0).

Step 2: Counting in Pascal's Triangle

To find the value of \( _7C_4 \), search the 7th row in the triangle which corresponds to the 8th row in conventional numbering (as we start from 0). Once in the correct row, count across for the 4th term (5th term in conventional counting).

Step 3: Find the Result

After locating the correct term within Pascal's Triangle, we find that the 5th term in the 8th row (representing \( _7C_4 \)) is 35.

Key Concepts

Understanding Binomial Expansion Coefficients

Binomial expansion coefficients are the numbers that are used to multiply the terms in a binomial expansion. A binomial expression is one that contains two terms, such as (a + b). When raised to a power, such as (a + b)^n, the expansion is expressed as a sum involving terms of a, b, and coefficients. These coefficients can be found easily using Pascal's Triangle.

To illustrate, let's consider an example of a binomial expression raised to the 7th power, (a + b)^7. The coefficients of each term, when expanded, correspond row by row to Pascal's Triangle. This is a result of the Binomial Theorem, which connects the coefficients to the concept of combinations. In essence, each coefficient in the expansion is the number of ways to pick items from a set without considering the order, which mathematically is expressed by combinations.

Calculating the _7C_4 Combination

The notation _7C_4, known as a combination, represents the number of ways to choose 4 items from a set of 7 without considering the order. This type of problem is commonplace in probability and combinatorics. Using Pascal's Triangle is a smart shortcut to calculate combinations without the lengthy calculations involving factorials.

To find _7C_4 using Pascal's Triangle, one would locate the 7th row and then identify the 5th entry (since we begin counting from 0). This correlates with the combination _7C_4 due to the properties of the triangle where each entry is essentially the combination of the row and the position. The value you find in the triangle is the number of distinct groups that can be formed, in this case, 35.

The Binomial Theorem Explained

The Binomial Theorem is a powerful tool in algebra that describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (a + b)^n into a sum that involves terms of the form a^k * b^(n-k) multiplied by a binomial coefficient. These coefficients are the ones we previously referred to as 'n choose k' or _nC_k, and they tell us how many combinations of k items can be selected from a set of n items.

The connection to Pascal's Triangle is that each coefficient from the binomial theorem is represented as an entry in the triangle. So, when looking at the binomial theorem from a combinatorial perspective, not only does it provide the expanded form of a powered binomial expression but it also gives insight into the number of possible combinations for each term's coefficients.