Problem 49 - Algebra for College Students

Question

Evaluate expression. \(\frac{14 !}{3 ! 11 !}\)

Step-by-Step Solution

Answer

364

Step 1: Understand the Factorial Notation

A factorial, denoted by an exclamation point (e.g., 14!), is the product of all positive integers less than or equal to that number. For example, 3! = 3 × 2 × 1 = 6.

Step 2: Write Down the Given Expression

The given expression is \(\frac{14 !}{3 ! 11 !}\).

Step 3: Expand Factorials in the Fraction

First, expand the factorials: \(\frac{14!}{3! \times 11!}\). Note that 14! = 14 × 13 × 12 × 11!.

Step 4: Cancel Common Factors

Since 11! appears in both the numerator and the denominator, it cancels out. Thus, the expression simplifies to: \(\frac{14 \times 13 \times 12 \times 11!}{3! \times 11!} = \frac{14 \times 13 \times 12}{3!}\).

Step 5: Compute the Remaining Expression

Next, compute 3!: \ 3! = 3 \times 2 \times 1 = 6 \. Now the expression simplifies to \(\frac{14 \times 13 \times 12}{6}\).

Step 6: Perform the Multiplication and Division

Multiply the numbers in the numerator: 14 × 13 = 182, then 182 × 12 = 2184. Now, divide by 6. \(\frac{2184}{6} = 364\).

Key Concepts

Factorial Notation

A factorial is a mathematical operation where you multiply a whole number by every whole number less than itself, down to 1. For instance, 5! (read as '5 factorial') means 5 × 4 × 3 × 2 × 1, which equals 120. This notation is symbolized by an exclamation mark (!). Factorial notation is often used in permutations and combinations, falling under the branch of combinatorics. It's handy to remember:

• 0! is always 1.
• Factorials grow very fast.
• They are extremely useful in counting problems.

In the problem \(\frac{14!}{3!11!}\), we have three factorials: 14!, 3!, and 11!. Each needs to be understood and handled accurately as we proceed with simplifying the expression.

Simplifying Fractions

Simplifying fractions is a crucial step in solving factorial-related problems. When you have a fraction like \(\frac{14!}{3!11!}\), the first task is to see which parts of the fraction can cancel out. This means identifying common factors in the numerator (top part) and the denominator (bottom part).
In this case, you can expand 14!:

• 14! = 14 × 13 × 12 × ... × 2 × 1

Noticing that 11! is a part of 14! (as in 14! = 14 × 13 × 12 × 11!), we can cancel out the 11! because it appears in both the numerator and the denominator. This leaves us with a simpler fraction:
\frac{14 \times 13 \times 12}{3!} .
When simplifying, always look for common factors that can be canceled. If the fraction is entirely simplified, perform any remaining operations on the remaining numbers.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. It's integral in fields like computer science, statistics, and even sports scheduling. In this specific problem, we use combinatorics to evaluate a binomial coefficient. The expression \(\frac{14!}{3!11!}\) is a way of counting how many ways we can choose 3 items from 14, also known as '14 choose 3' or combinatoric notation. This simplifies our problem:

• Understanding the arrangement and grouping in combinatorics can turn complex problems into manageable calculations.
• Always simplify factorial expressions for easier computation.

By canceling out the factorials, we transform combinatorial problems into basic arithmetic operations, making them easier to solve and understand.