What is " zero factorial "? One(1)? Absolutely correct !! Now why is that so? Not having the answer. Check it
0! = 1 for reasons that are similar to why x^0 is equals to 1. Both are defined that way.
1) You cannot reason that x^0 = 1 by thinking of the meaning of powers of x as "repeated multiplications of x" because you cannot multiply x zero times.
• x^0 = 1 is defined in order to make the laws of exponents work even when the exponents can no longer be thought of as repeated multiplication. The formula is :
(x^a) (x^b) = x ^ (a+b)
• For example, (x^73)(x^27) = x^100 because you can add exponents. You're not going to multiply x 73 times and 27 time then multiplying them toghether. Are you?
• In the same way (x^0)(x^2) should be equal to x^2 by adding exponents. But that means that x^0 must be 1 because when you multiply x^2 by it, the result is still x^2. Only x^0 = 1 makes sense here.
2) Similarly, you cannot reason out 0! just in terms of the meaning of factorial because you cannot multiply all the numbers from 0 DOWN TO 1 to get 1. (Not making sense huh!)
• We know the formula to find the number of ways we can choose "r" things out of "n" things which is
nCr = n! / r! (n-r)!
• Now suppose that there are 2 people and "everybody shakes hands with everybody else." How many handshakes will be there? Obviously one and only one!
• Now put n = 2 (total 2 people) and
c = 2 (2 people are shaking
their hands out of 2)
in the formula we get,
nCr = 2! / (2! 0!)
But we know nCr=1,
1 = 2! / (2! 0!)
Say 0! = x,
1 = 2! / (2! x)
Simplifying we get x=1 that is 0! = 1.
There are many other reasons why the value of 0! = 1 but the reasons above are the most sense making. Mathematicians have pre defined many such logically related assumptions in order to make math more easy and more consistent. Also to remain consistent with other mathematics worldwide.
That's it! Tell me if you have something more information about this topic or having any doubts.