Problem 4 - Precalculus

Question

Evaluate the given binomial coefficient. $$\left(\begin{array}{c}11 \\\1\end{array}\right)$$

Step-by-Step Solution

Answer

The value of the binomial coefficient \(\left(\begin{array}{c}11 \ 1\end{array}\right)\) is 11.

Step 1: Identify n and k in the binomial coefficient

The given expression is \(\left(\begin{array}{c}11 \ 1\end{array}\right)\). Here, n=11 and k=1.

Step 2: Apply Binomial Coefficient Formula

The formula for calculating binomial coefficient is \(\left(\begin{array}{c}n \ k\end{array}\right) = \frac{n!}{k!(n-k)!}\). Substituting the values of n and k into the formula, we get \[\frac{11!}{1!(11-1)!} = \frac{11!}{1! \cdot 10!}\]

Step 3: Calculate Factorials

11! means multiplying all positive whole numbers from 1 to 11, while 1! and 10! represents multiplying all numbers from 1 to 1 and 1 to 10 respectively. We calculate this as: \(11! = 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1, 1! = 1 \text{ and } 10! = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\)

Step 4: Simplify the Expression

The 10 factorial in the numerator and the denominator will cancel out. Thus, the expression becomes: \(\frac{11}{1} = 11\)

Key Concepts

Understanding Factorials

Factorials are a fundamental concept in mathematics, symbolized by the exclamation mark (!). The factorial of a number, say \( n \), denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Some important points about factorials are:

• It’s used extensively in permutations and combinations.
• \( 0! = 1 \) as a special case by definition.
• Factorials grow very quickly with larger numbers.

In the context of binomial coefficients, factorials help in calculating combinations, which are ways to pick items from a larger pool.

Combinatorics and Binomial Coefficients

Combinatorics is the branch of mathematics dealing with combinations and permutations of objects. A key part of combinatorics is understanding binomial coefficients, used to find combinations without considering the order.

The binomial coefficient is denoted by \( \binom{n}{k} \) and calculated as \( \frac{n!}{k!(n-k)!} \). Here's how it works:

• \( n \) represents the total number of items.
• \( k \) is the number of items to choose.
• The formula accounts for all possible combinations.

In our example, \( \binom{11}{1} \), it shows how to select 1 item from 11.

The result, calculated through factorials, is 11, as only one choice from many results in many solutions.

Exploring Pascal's Triangle

Pascal's Triangle is a geometric arrangement of binomial coefficients. Each number in the triangle is the sum of the two numbers directly above it.

Here's how it connects to binomial coefficients:

• The nth row corresponds to the coefficients in the expansion of \( (a+b)^{n} \).
• The entries are the binomial coefficients \( \binom{n}{k} \).
• It provides a visual way to compute combinations quickly.

For the example given, \( \binom{11}{1} \), you would find it in the second position of the 11th row of Pascal's Triangle. It visually represents how we choose 1 item from 11, reinforcing the calculated value of 11.