Question

Evaluate each expression. $$ \frac{14 !}{5 ! 9 !} $$

   Evaluate each expression.
$$
\frac{14 !}{5 ! 9 !}
$$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 11, Problem 34

Instant Answer

Solved by Expert Joshua Eastwood

03/15/2020

Step 1

We can write this out as $\frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(5 \times 4 \times 3 \times 2 \times 1) \times (9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 Show more…

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Evaluate each expression. $$ \frac{14 !}{5 ! 9 !} $$

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Key Concepts

Factorial

A factorial, denoted by n!, is the product of all positive integers from 1 to n. It is a fundamental concept used in many areas of mathematics, especially in counting, probability, and series expansions. Factorials provide a convenient way to represent the number of ways to arrange n distinct objects.

Binomial Coefficient

The binomial coefficient is represented as n choose k and is defined as n!/(k!(n-k)!). It calculates the number of ways to choose k elements from a set of n distinct elements without considering the order. This concept is central to the study of combinatorics and appears in the binomial theorem.

Simplification of Factorial Expressions

Simplifying expressions involving factorials often involves canceling common terms in the numerator and denominator. By recognizing the overlapping parts of factorials, one can reduce complex expressions to simpler forms, which makes evaluating them much more manageable.

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