Ms. Q gave Grogg the following problem: "License plates in Aopslandia consist of six upper-case letters. For example, two possible Aopslandian license plates are $ABCDEF$ and $AAAOPS.$ No two license plates are the same. How many possible Aopslandian license plates are there which contain at least four $A$'s?" Grogg got the answer $\binom{6}{4} \cdot 26^2,$ but Ms. Q told him this was the wrong answer! (a) How did Grogg arrive at his answer?
(b) Why is Grogg's answer wrong? Should the correct answer be smaller or larger than Grogg's answer (and why)?
(c) Write a solution to Ms. Q's problem, explaining in complete sentences what the correct answer to the problem should be and why.