Question

Which of the following expressions is equivalent to \(\frac{12 !}{15 !} ?\) $$ \begin{aligned} & \text { } \frac{\frac{4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1}}{\frac{1}{15 \times 14 \times 13}} \\ & \frac{4}{5} \end{aligned} $$ I don't know.

Short answer

None of the given options are equivalent to \( \frac{12!}{15 !} \). The simplified expression for \( \frac{12!}{15 !} \) is \( \frac{1}{15 \times 14 \times 13} \).

Step-by-step solution

Compare the factorials

We're looking at the expression \( \frac{12 !}{15 !} \). Since 12! is 12 x 11 x 10 x .... x 2 x 1 and 15! is 15 x 14 x 13 x 12 x 11 x 10 x...x 2 x 1, we see that 12! is a part of 15!.

Simplify the expressions

Since 12! is part of 15!, we can simplify by eliminating the common terms from the numerator and denominator of the expression as follows: \( \frac{12 !}{15 !} = \frac{1}{15 \times 14 \times 13} \).

Compare the given options

Comparing the obtained expression with the given options, it is clear that None of the given options are equivalent to \( \frac{12!}{15 !} \).

Key concepts

Mathematical Expressions

In mathematics, expressions are combinations of symbols that represent a value or relationship. A mathematical expression can be as simple as a single number or variable, or as complex as a series of operations. In our exercise, we deal with a term \[\frac{12!}{15!}\]that uses factorials. A factorial, denoted by an exclamation mark such as in '12!', refers to the product of all positive integers up to a specified number. This means:

• 12! = 12 x 11 x 10 x ... x 2 x 1
• 15! = 15 x 14 x 13 x 12 x 11 x 10 x ... x 2 x 1

When writing mathematical expressions such as this, it's crucial to detail out terms to simplify or restructure them as needed.

Simplification

Simplification is the process of reducing complex mathematical expressions to their simplest form without changing their value. In our problem, the simplification lies in recognizing the overlaps within the factorial expressions:
\[\frac{12!}{15!}\]
Since 12! is a factor in 15!, simplification involves canceling out the common terms. By removing the identical sequence from both the numerator and the denominator, we are left with:\[\frac{1}{15 \times 14 \times 13}\]
Simplifying expressions is not only useful for easier calculations but also for comparing equations or expressions more efficiently. Always look to cancel out common factors as a primary simplification method.

Numerator and Denominator

The terms numerator and denominator are fundamental in understanding fractions. Every fraction consists of a numerator, the top number, and a denominator, the bottom number, which indicate a part-to-whole relationship.
In the problem, consider\[\frac{12!}{15!}\]- the `12!` is the numerator, and the `15!` is the denominator.
The comparison of these two parts allows us to simplify or evaluate the fraction. By removing the overlapping factors of `12!` in the denominator, we focus the operations on simplifying the ratio of these expressions. Understanding this relationship becomes crucial for more complex calculations like those you'll find in combinatorics.

Combinatorics

Combinatorics is a field of mathematics concerned with counting, arrangement, and combination of objects. Factorials appear frequently in combinatorial calculations because they denote permutations - the number of ways to arrange a set number of items.
For instance, the expression in our task, \[\frac{12!}{15!}\]can relate to a problem in combinatorics where you choose a subset of 12 out of a larger set of 15 items, and observe the total arrangement possibilities (though that's a broad connection here).
Combinatorics often requires simplification and understanding of expressions involving factorials. It helps develop solutions for problems involving order and structure, so having a solid grasp of simplifying mathematical expressions and managing numerators and denominators is extremely beneficial.