Let us introduce the notation – we denote the product of all natural numbers from 1 to \(n\) by \(n!\). For example, \(5!=1\times2\times3\times4\times5=120\).
a) Prove that the product of any three consecutive natural numbers is divisible by \(3!=6\).
b) What about the product of any four consecutive natural numbers? Is it always divisible by 4!=24?
Is it possible for \(n!\) to be written as \(2015000\dots 000\), where the number of 0’s at the end can be arbitrary?
The numbers \(1, 2, \dots , 9\) are divided into three groups. Prove that the product of the numbers in one of the groups will always be no less than 72.
The number of permutations of a set of \(n\) elements is denoted by \(P_n\).
Prove the equality \(P_n = n!\).