Factorials

I would like to dive straight into the concept of factorials. It is a simple mathematical operation used to find the total ways in which certain entities (numbers, people, objects, etc.) can be arranged in a straight line. All numerical factorials are represented by a number followed by an exclamation mark.

$$ \text {For example, 3 factorial is 3!} \space 4!, 5!, 100!, n! $$

Let us take a few examples and slowly build our understanding of factorials.

1. How many ways are there in which one entity can be arranged
   • Only one way, since there is no other entity to swap the position with. (Refer to entity as mentioned at the beginning of the article.)
2. If David and Sam are to be seated on two chairs, how many ways are possible to do so?
   • $$ 2 \times 1 = 2 $$
   • $$ \text {(Chair 1 – David; Chair 2 – Sam)} \space and \space \text {(Chair 1 – Sam; Chair 2 – David)} $$
   • Hence,
3. There are three numbers (1, 2, 3). How many unique three-digit numbers can we get? (note: the digits must not be repeated in a number.)
   • $$ 123, 132, 231, 213, 321, 312 $$
   • Keep every number at the 100s place. Then we are left with two choices about how we can arrange the remaining two numbers.
   • e.g., for digits ABC, if we place C at the 100s place, then we can write AB and BA after C, which gives us two unique numbers, CAB & CBA.
   • This way, we can write two numbers for fixing one number at the hundreds place. Hence there are ways (or unique numbers).
   • Hence,
4. In our final example, we will see how many unique four-digit numbers we can create using four different numbers, with the condition of non-repeatability of numbers as it is (the digits must not be repeated in a number).
   • For each number at the 1000s place, we can arrange the other three numbers in six different ways, as seen in the above example.
   • Therefore, we can create

Now that we have established the idea of factorials, let us crunch some more numbers and strengthen this concept, as it has applications in daily life situations, like arrangements of objects and people in various ways, engineering, differential equations, and various other fields.

$$ 6! = 6\times5\times4\times3\times2\times1 \newline = (6\times5)\times(4\times3)\times(2\times1) \newline = 30\times(12\times2) = 30\times 24 = 720 $$

$$ 5! = 5\times4\times3\times2\times1 \newline = (5\times4)\times(3\times2)\times1 \newline = 20\times6\times1 = 120 $$

$$ \text {Note: You can choose to put brackets on any numbers as per your convenience,} \newline \text {since it is numerical multiplication and is not affected by position of brackets.}$$

I would like to conclude the article here, having explained the way to do factorials and its applications in various fields, including but not limited to mathematics, thermodynamics, quantum mechanics, statistics, and finance. You can extend this exercise for finding factorials of numbers from 1 to 20, as these are manageable numbers. You can also go further, but you will realize that the calculation required is lengthy (although not impossible) and time-consuming, and very big numbers have to be dealt with. Keep learning, stay curious.