Answer the following questions about a scheduling system for assigning teachers to office hours. Within the system, every single teacher and every single office hour time is assigned an integer ID. These IDs start with 1 and increment by 1. That is, if there is a teacher with ID equal to 6, there must be teachers with IDs equal to 1, 2, 3, 4 and 5. The same restriction applies to the office hour IDs. Note that any office hour not assigned a teacher will be covered by a substitute teacher. (a) Consider the teachers with IDs ranging from a to b inclusive, where a ≤ b, and consider the office hour slots with IDs ranging from c to d inclusive, where c ≤ d. How many distinct functions for assigning these teachers to office hours are there? (The teachers are the domain and the office hours are the codomain.)
(b) Consider the teachers and office hours with IDs ranging from a to b inclusive, where a ≤ b. How many distinct functions for assigning teachers to office hours are there, such that every teacher is assigned an office hour with an ID that is less than or equal to their ID?