Problem 22 - Beginning and Intermediate Algebra

Question

Evaluate each binomial coefficient. $$\left(\begin{array}{c}11 \\\8\end{array}\right)$$

Step-by-Step Solution

Answer

The binomial coefficient \(\binom{11}{8}\) can be evaluated as \(\frac{11!}{8!(11-8)!} = \frac{39916800}{40320 \times 6} = 165\). Thus, \(\binom{11}{8} = 165\).

Step 1: Write down the formula for a binomial coefficient

Recall the formula for a binomial coefficient: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Step 2: Substitute n and k values in the formula

We have n = 11 and k = 8, so we now substitute these values into the formula: \[\binom{11}{8} = \frac{11!}{8!(11-8)!}\]

Step 3: Evaluate factorial expressions

Compute the factorials in the expression: \[11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39916800\] \[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\] \[3! = 3 \times 2 \times 1 = 6\]

Step 4: Compute the binomial coefficient

Plug the computed factorials back into the expression and evaluate the binomial coefficient: \[\binom{11}{8} = \frac{39916800}{40320 \times 6} = \frac{39916800}{241920} = 165\] The binomial coefficient \(\binom{11}{8}\) is equal to 165.

Key Concepts

Understanding Factorials

A factorial of a number is a mathematical operation that involves multiplying a series of descending natural numbers. When we talk about the factorial of a number, it is denoted by that number followed by an exclamation mark, such as 5!.
Here's how it works:

• For a number like 5, the factorial is calculated as: 5! = 5 × 4 × 3 × 2 × 1 = 120.
• The factorial of 0 is defined to be 1.

Factorials are important in the calculation of binomial coefficients, which you'll encounter in various areas of combinatorics and algebra. In the context of a binomial coefficient, factorials simplify the counting of combinations, making it easier to determine the number of ways to choose items from a set.
In our exercise, we calculated 11!, 8!, and 3! as part of finding the binomial coefficient.

Combinatorics and Binomial Coefficients

Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting. It provides essential tools for counting and arranging objects in a systematic way.
A key concept in combinatorics is the binomial coefficient, represented as \( \binom{n}{k} \). This represents the number of ways to choose \( k \) items from \( n \) total items without regard to order.

• The formula is: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

This formula uses factorials to account for all arrangements and divides by \( k! \) and \( (n-k)! \) to eliminate over-counting rearrangements.
In the exercise, we computed \( \binom{11}{8} \) which means choosing 8 items from a set of 11 items. The calculated result was 165, representing the number of combinations possible.

Algebraic Perspectives on Binomial Coefficients

Algebra involves the manipulation of symbols and rules to express relationships and solve equations. Binomial coefficients hold a significant role in algebra, especially within expansions and series.
When expanding expressions like \( (a+b)^n \), binomial coefficients dictate the coefficients in each term of the expanded expression. This is part of the binomial theorem.

• For instance, the expansion of \( (a+b)^3 \) is \( a^3 + 3a^2b + 3ab^2 + b^3 \). Coefficients 1, 3, 3, 1 correspond to \( \binom{3}{0}, \binom{3}{1}, \binom{3}{2}, \binom{3}{3} \).

Algebraically, computing binomial coefficients is essential in understanding polynomial identities and solving combinatorial problems expressed as algebraic equations.
In our scenario, calculating \( \binom{11}{8} \) helps understand how often certain polynomial terms will appear when manipulating algebraic expressions.