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Factorial notation is a mathematical concept represented by an exclamation mark '!' following a number. For any positive integer 'n', the factorial notation 'n!' represents the product of all natural numbers from 1 to 'n'.
For example:

• 1! = 1
• 2! = 2 × 1 = 2
• 3! = 3 × 2 × 1 = 6
• 4! = 4 × 3 × 2 × 1 = 24

Factorials grow very fast—much faster than simple addition or multiplication. This makes them interesting and useful in various mathematical calculations, such as permutations and combinations.

Multiplication is one of the fundamental arithmetic operations. When dealing with factorials, you multiply a series of decreasing integers.
To understand this better, let's consider the example of finding 4! Follow these steps to perform the necessary multiplications:

• Step 1: Begin with the largest number, 4.
• Step 2: Multiply it by the next largest number, 3. Result = 12.
• Step 3: Multiply the result by the next number, 2. Result = 24.
• Step 4: Multiply by the smallest number, 1 (which does not change the result). Final Result = 24.

You can represent these steps using the equation:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24. \] Multiplying in this sequential manner ensures that we account for each integer in the factorial chain.

Simplification involves breaking down an expression into its most basic form. Factorials might look complicated at first, but the process to simplify them is straightforward.
First, write down all the steps, as seen in the definition of the factorial notation. This involves listing all the multiplicative components.
Here is the sequence for 4!:
\[4 \times 3 \times 2 \times 1 = 24. \]
Simplifying the factorial, 4!, means performing the multiplications in sequence:
\[4 \times 3 = 12 \]
\[12 \times 2 = 24 \]
\[24 \times 1 = 24. \] As a final step, make sure your simplified result matches the definition. The goal is to make the expression easy to handle and understand without altering its value.