Problem 20 - Beginning and Intermediate Algebra

Question

Evaluate each binomial coefficient. $$\left(\begin{array}{l}9 \\\3\end{array}\right)$$

Step-by-Step Solution

Answer

The binomial coefficient \(\binom{9}{3}\) is calculated using the combination formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). After computing the factorials and substituting the values, we find that \(\binom{9}{3} = \frac{362880}{4320} = 84\).

Step 1: Calculate the Factorials

Calculate the factorial values for 9!, 3!, and (9-3)!. \(9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880\) \(3! = 3 \times 2 \times 1 = 6\) \(6! = (9-3)! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

Step 2: Apply the Combination Formula

Now, plug the factorial values into the combination formula: \(\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{362880}{6 \times 720}\)

Step 3: Simplify the Fraction

Simplify the fraction to find the value of the binomial coefficient: \(\binom{9}{3} = \frac{362880}{4320} = 84\) So, the value of the binomial coefficient \(\binom{9}{3}\) is 84.

Key Concepts

Factorial

Factorial is a fundamental concept in mathematics, specifically in permutations and combinations. It is denoted by the symbol "!" (exclamation point) and involves multiplying a series of descending natural numbers.
For example, the factorial of 5 (written as 5!) is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

The concept of factorial is crucial in combinatorial mathematics as it helps in determining the number of ways to arrange or choose items.
More precisely, factorials are used to calculate permutations, which is all about arranging things in order, and combinations, which relates to selecting items without regard to order.
When evaluating a binomial coefficient, like \((\binom{9}{3})\), you calculate the factorials involved in the numerator and the denominator to solve the problem.
Understanding factorials lays the foundation for grasping more complex mathematical concepts involved in probability and algebra.

Combination Formula

The combination formula is a cornerstone in calculating binomial coefficients. It allows us to find the number of ways to select a group of objects from a larger set where the order does not matter.
The standard formula for combinations is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, \(n\) is the total number of items, and \(r\) is the number of items to choose.
In our given problem, \(\binom{9}{3}\), we used this formula by plugging in the values:

• \(n = 9\)
• \(r = 3\)

We calculated \(9!\), \(3!\), and \((9-3)!\) to arrive at the solution.
This formula helps simplify calculations for larger numbers, making it easier to determine combinations without listing all possibilities.

Simplify Fractions

Simplifying fractions is a necessary step in solving many mathematical problems, including evaluating binomial coefficients. Once you have the fractions from applying the combination formula, simplifying them makes it easier to find the exact value.
To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).
When you have a fraction like \(\frac{362880}{4320}\) from our binomial coefficient example, this step involves finding the GCD of 362880 and 4320 and dividing both parts of the fraction by this number.

• The simplified form will ultimately provide the numerical value for the binomial coefficient.

In our problem, simplifying \(\frac{362880}{4320}\) results in the binomial coefficient value of 84.
Simplifying fractions is an essential skill in mathematics, allowing for more manageable and meaningful calculations.