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Factorial notation is a mathematical symbol used to indicate the product of all positive integers up to a given non-negative integer, denoted by an exclamation mark (!). For instance, if we have the number 4, its factorial (written as 4!) will be calculated as: 4! = 4 × 3 × 2 × 1 = 24. This notation becomes particularly useful in various areas of mathematics, including permutations, combinations, and algebra.

It's important to note that 0! (0 factorial) is defined to be 1. This might seem confusing at first, but it makes many mathematical formulas work correctly. So, even though 0 has no positive integers less than it, its factorial value is 1 by definition.

Simplifying fractions means reducing a fraction to its lowest terms. This process involves dividing both the numerator (top part) and the denominator (bottom part) of the fraction by their greatest common divisor (GCD). Let's break it down step by step:

If we start with a fraction, like \(\frac{4!}{0! \times 4!}\), we can begin by calculating each factorial separately. From the earlier calculation, we know that 4! = 24 and 0! = 1. So, the fraction becomes \(\frac{24}{1 \times 24}\).

Next, we simplify this fraction by dividing the numerator and the denominator by their GCD. Here, both the numerator and the denominator have a GCD of 24. Simplifying gives us: \(\frac{24}{24} = 1\). Therefore, the simplified value of the given fraction is 1.

In mathematics, a non-negative integer is any integer that is not less than zero. In other words, it includes all positive integers (1, 2, 3, ...) as well as zero (0). Negative numbers are not part of non-negative integers.

The concept of non-negative integers is important when dealing with factorials because factorial functions are only defined for these numbers. For example, we can find the factorial of 0, 1, 2, etc., but not -1 or any other negative number.

Understanding non-negative integers helps in various mathematical operations and ensures that the solutions remain valid within the defined domain of the problem.