Factorial Rules

Factorial: Types & Uses


1.     What is Factorial ?


Factorial of a number n is the product of all natural numbers from 1 to n.


n! = n × (n-1) × (n-2) ×...× 1


Ex: 5! = 5 × 4 × 3 × 2 × 1 = 120


2.     Types of Factorial & Tricks


i.   Positive Integer Factorial : Applies to natural numbers only (1, 2, 3, 4…)


       Formula: n! = n × (n-1)×...×1


      Ex: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720


ii.    Zero Factorial : Special case in mathematics


       Although 0 has no number to multiply, by definition it is still counted as 1.


      Ex: 0! =1, In permutations & combinations, when nothing is left to arrange, there is 1 possible way (empty arrangement).


iii.    Double Factorial (n!!) : Multiply every 2nd number (skip by one each time)


      Ex: 7!! = 7 × 5 × 3 × 1 = 105


iv.   Even Double Factorial : When the starting number is even, multiply only even numbers down to 2


      Ex: 8!! = 8 × 6 × 4 × 2 = 384


v.    Odd Double Factorial : When the starting number is odd, multiply only odd numbers down to 1


     Ex: 9!! = 9 × 7 × 5 × 3 × 1 = 945


vi.   Gamma Function (Extension of Factorial) : Used when factorial is needed for fractions or decimals



  • It is a special advanced function

  • Formula: n! = r(n+1)


     Ex: 4! = r(5)


Factorial



  • 1! : 1 = 1

  • 2! : 2 × 1 =2

  • 3! : 3 × 2 × 1 = 6

  • 4! : 4 × 3 × 2 × 1 = 24

  • 5! : 5 × 4 × 3 × 2 × 1 = 120

  • 6! : 6 × 5 × 4 × 3 × 2 × 1 = 720

  • 7! : 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

  • 8! : 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320

  • 9! : 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880

  • 10! : 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800


Trick : Each factorial is just previous one multiplied by next number: 6! = 720 , 7! = 6! × 7 = 720×7 = 5040, 8! = 7! × 8 = 5040 × 8 = 40320


Best Short Trick


If letters repeat: Divide by repeated letter factorials


Format: Total arrangements = n!/(p!)(q!)(r!)


where, n = total letters  and p,q,r = factorial of repeated letters 


Rule (When letters do NOT repeat) :


If all letters are distinct: Number of permutations = Total letters!


Meaning: Just take factorial of the total number of letters.


Quick Trick Summary (Permutation)


1st :  All letters different


Trick: Just do n!, Reason: No repetition → no division


Ex: S,M,I,L,E (5 distinct letters)


5! = 120


2nd : Some letters repeat (only one letter repeats)


Trick: Do n! ÷ (factorial of repeats), Reason: Divide by repeated letter count to remove duplicates


Ex: I,N,D,I,A (I repeats 2 times)


5! / 2! = 60


3rd: Case: Multiple letters repeat


Trick: Divide by each repeated letter’s factorial, Reason: More than one letter repeats → divide by all repetition factorials


Ex: C,O,M,M,I,T,T,E,E (M = 2 times, T = 2 times, E = 2 times)


9! / 2!×2!×2!, Calculate factorials 9! = 362880, 2! = 2


= 362880 / 8


= 45360


Super Shortcut



  • All different → n!

  • One letter repeats → n! ÷ (repeat!)

  • Several letters repeat → n! ÷ (each repeat factorial multiplied)