in additionx ** ln( xx there are number of useful inequalities relating to these functions (which derive from these definitionsproposition if - <ln( < + ex proposition for < <ex < - proposition for any two positive real numbers and <ex < + / integer functions and relations the "floorand "ceilingfunctions are defined respectively as follows xthe largest integer less than or equal to xthe smallest integer greater than or equal to the modulo operator is defined for integers > and as mod the factorial function is defined as ( ) the binomial coefficient is nk!( ) which is equal to the number of different combinations one can define by choosing different items from collection of items (where the order does not matterthe name "binomial coefficientderives from the binomial expansionn - ab ( bk= we also have the following relationships