Answer using discrete mathematics to solve below:

A company has $15$ employees divided into three departments: Research $(6$ employees$),$ Development $(5$ employees$),$ and Testing $(4$ employees$).$ The company needs to form specialized project teams under the following conditions:

$$ Each team must consist of $5$ employees with at least $2$ employees from Research and at least $1$ employee each from Development and Testing.

$$ Within each team, the employees are assigned distinct roles$($e$.$g$.,$Lead,Analyst,Tester,Developer, and Coordinator$)($Hint: the arrangement of the employees matters$)$

Given these conditions, how many ways are there to select the team members to satisfy the departmental requirements for a single team?