Problem 25 - Beginning and Intermediate Algebra

Question

Evaluate each binomial coefficient. $$\left(\begin{array}{l}6 \\\1\end{array}\right)$$

Step-by-Step Solution

Answer

The value of the binomial coefficient \(\binom{6}{1}\) is 6, calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) and substituting the values \(n = 6\) and \(k = 1\).

Step 1: Write down the formula for binomial coefficient

In order to find the binomial coefficient \(\binom{6}{1}\), we will use the general formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

Step 2: Plug in the values for n and k

Now, we will plug in the values for n and k in the formula. In our case, \(n = 6\) and \(k = 1\), so the binomial coefficient becomes: \[\binom{6}{1} = \frac{6!}{1!(6-1)!}\]

Step 3: Calculate the factorials

Before we can perform the division, we need to calculate the factorials. - \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\) - \(1! = 1\) - \((6-1)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) Now, substitute the factorials back into the formula: \[\binom{6}{1} = \frac{720}{1 \times 120}\]

Step 4: Simplify the expression and find the answer

Finally, we will simplify the expression to find the value of the binomial coefficient. \[\binom{6}{1} = \frac{720}{120} = 6\] Therefore, the value of the binomial coefficient \(\binom{6}{1}\) is 6.

Key Concepts

Understanding Factorials

Factorials are a fundamental concept in mathematics, especially in combinatorics and algebra. The factorial of a number, denoted by an exclamation point (e.g., \( n! \)), is found by multiplying all positive integers up to that number. For instance, \( 6! \) is calculated as \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \).

Here are a few key points about factorials:

• \( 0! \) is defined as 1.
• Factorials grow very quickly with larger numbers.
• They are integral to calculating permutations and combinations.

Factorials help simplify the calculations involved in binomial coefficients, as they provide a way to express product sequences in a compact form.

Exploring Combinatorics

Combinatorics deals with counting, arrangement, and combination of objects. One of the essential tools in combinatorics is the binomial coefficient, which helps determine how many ways you can choose \( k \) objects from \( n \) without caring about order.

The binomial coefficient \( \binom{n}{k} \) is given by the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] This formula is fundamental in calculating permutations and combinations.

• For the expression \( \binom{6}{1} \), it represents selecting one element from six, which is simply 6.
• It shows symmetry as \( \binom{n}{k} = \binom{n}{n-k} \).
• Used extensively in probability theory and statistical models.

Combinatorics makes it easier to understand arrangements and choices, providing clarity on how elements relate to one another.

Algebraic Concepts in Binomial Coefficients

Algebra often uses binomial coefficients to simplify expressions, particularly in binomial expansion. The rules of algebra allow us to manipulate these coefficients for diverse applications, including solving equations and understanding polynomial behavior.

Binomial coefficients appear in Pascal's Triangle, where each coefficient is the sum of the two directly above it. This relationship is a classic demonstration of algebraic patterns.

Key points include:

• The connection between algebra and combinatorics through the binomial theorem.
• Understanding how variables interact in algebraic expressions using binomial coefficients.
• The ability to break down complex algebraic expressions into manageable parts using known formulas.

By incorporating algebra with binomial coefficients, we can deepen our comprehension of mathematical structures and enhance problem-solving skills.