Answer: A

Small Hint:

The number of assignments is $\dfrac{16!}{(4!)^4\, 4!} \cdot 12^4,$ where each committee contributes $4 \cdot 3 = 12$ chair-and-secretary choices

Big Hint:

Track only powers of $3:$ $16!$ gives $3^6,$ the denominator $(4!)^4\,4!$ gives $3^5,$ and $12^4$ gives $3^4$

Solution:

The number of ways to split $16$ people into $4$ indistinguishable groups of $4$ is $\dfrac{16!}{(4!)^4\, 4!}.$ Each committee then chooses a chairperson and a secretary in $4 \cdot 3 = 12$ ways, contributing $12^4.$ So the total is $\dfrac{16!}{(4!)^4\,4!}\cdot 12^4.$

Counting factors of $3:$ $16!$ contributes $\lfloor \frac{16}{3}\rfloor + \lfloor \frac{16}{9}\rfloor = 6.$ The denominator $(4!)^4\,4!$ contributes $4 + 1 = 5.$ And $12^4 = (2^2\cdot 3)^4$ contributes $4.$ Thus $r = 6 - 5 + 4 = 5.$

Thus, the correct answer is A.