A note on factorials

Factorials are very simple things. They're just products, indicated by an exclamation mark. For instance, "four factorial" is written as "4!" and means 1×2×3×4 = 24. In general, n! ("enn factorial") means the product of all the natural numbers from 1 to n; that is, n! = 1×2×3×...×n.
The most common factorials are:
1! = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
Did you notice any pattern by any chance? 6! = 6 x 5!. Likewise 4! = 4 x 3!. Isn’t that an interesting observation? In general, we can write:
(n + 1)! = (n + 1) x n!.

What if we put n = 0 here?
1! = 1 x 0!
Or, 0! = 1
In fact I have seen many otherwise intelligent people making the fundamental error of assuming 0! = 0.
We can probably formulate a question like: If A and B are numbers such that A! = B! and A ≠ B, then how many such ordered pair of (A, B) exists?
Obviously two, as A and B can only have values 0 and 1 since only for these values we have 0! = 1!.

What if we have to simplify the expression given below?
p = 1 + 2x2! + 3x3! + ….10x10!

How do we approach these problems?
See that the nth term here, say Tn = nxn!.
But we have learnt that (n+1)n! = (n+1)!.
So, let’s now write nxn! as (n + 1 –1)xn! = (n+1)n! –n!.
Or, Tn = (n+1)! – n! ; T1 = 2! – 1! ; T2 = 3! – 2! and so on.
Obviously every other term will get cancelled in the sum and we would be left with 11! – 1!

By the way we just solved a 2 mark question in a very recent CAT. Half a dozen of such questions and the cut-off needed to clear CAT would be through.