Problem 16 - Precalculus with Limits

Question

In Exercises 15 - 18, evaluate using Pascals Triangle. \( \dbinom{9}{6} \)

Step-by-Step Solution

Answer

The value of \( \dbinom{9}{6} \) is '84'.

Step 1: Recalling Pascal's Triangle

Pascal's Triangle is an array of binomial coefficients. It is a triangular array constructed by summing adjacent elements in preceding rows. The triangle starts with a '1' at the top. The second row is two '1's, the third row is '1, 2, 1', and so on. To reach the 9th row (keeping in mind that the number of row is equal to the number of 'n' in the binomial), it must continue down the triangle.

Step 2: Locating Required Coefficient in Pascal's Triangle

When the 9th row of Pascal's Triangle is reached, the coefficients will be: '1, 8, 28, 56, 70, 56, 28, 8, 1'. Each number in this row is a binomial coefficient for \( \dbinom{9}{k} \), where 'k' stands for the position in the row, starting with 0. To find \( \dbinom{9}{6} \), count to the 7th spot in the 9th row (because it starts from 0).

Step 3: Providing the Solution

The 7th element in the 9th row of the Pascal's Triangle is '84'. So, \( \dbinom{9}{6} \) = '84'.

Key Concepts

Understanding Binomial Coefficients

Binomial coefficients are the numbers found in Pascal's Triangle. They represent the coefficients in the expansion of a binomial expression like \((a + b)^n\). These coefficients are denoted by \( \binom{n}{k} \), where 'n' is the row number, and 'k' is the position within that row, starting from zero.
For example, \( \binom{9}{6} \) indicates we want the coefficient from expanding \((a + b)^9\), specifically the term involving \(a^3b^6\) if considering a full binomial expansion.
Binomial coefficients can also be calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \(n!\) denotes the factorial of 'n', which is the product of all positive integers up to 'n'. This formula is incredibly useful in combinatorics and dealing with large expansions.

Exploring Combinatorics

Combinatorics is the field of mathematics focused on counting, arranging, and finding patterns among sets of elements. It deals with questions like "How many ways can we arrange or select items?".
Binomial coefficients are a key part of combinatorics. They count the number of ways to choose 'k' elements from a set of 'n' elements, without regard to order.

• Permutations: These are arrangements where the order does matter.
• Combinations: These involve selections where order doesn't matter, which is where binomial coefficients often come in.

By using Pascal's Triangle, one can quickly find the number of combinations possible, such as choosing 6 items from a pool of 9 using \( \binom{9}{6} \).
Understanding these concepts helps in solving problems related to probability, algebra, and statistical calculations.

Introduction to the Binomial Theorem

The Binomial Theorem provides a way to expand expressions raised to a power, like \((a + b)^n\). Using binomial coefficients, the theorem helps us expand any power of a binomial into a sum of terms:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
Each term in this expansion corresponds to a binomial coefficient \( \binom{n}{k} \), multiplied by the terms \(a\) and \(b\) raised to the appropriate powers.
This theorem is not only important in algebra but also in probability and calculus. It allows us to see the connection between algebraic expressions and combinatorial patterns.
For example, using the Binomial Theorem, one can verify that the 6th coefficient in \((a + b)^9\) is indeed \(84\), matching what we found using Pascal’s Triangle for \( \binom{9}{6} \).