I solved it!!! And here's my answer! (Btw, I didn't go to the link when solving it.)
Grogg's Reasoning
a) The Answer Grogg gave is C(6,4) multiply by the 26 square. Grogg's reasoning must have been: Because we are repeating the letter "A" four times we are overcounting. Then, he multiplied his answer by the square of 26 as the word is still not complete. There are 26 letters in the alphabet and the next letters could be any letter in the Alphabet so we multiply all the choices.
What Grogg did Wrong
b) Grogg's answer is wrong because of multiple reasons. One reason is, Grogg, multiplied by 26 square. 26 square means that selecting A, is possible. However, we cannot choose A again in this specific case as this case is for only 4 A's. The third reason why Grogg's Answer is wrong is he only attempted to answer the first Case. There are 3 cases here.
Case 1: Only 4 A's. The equation is C(6,4) x (25 +25 x 24 x 2!).
Case 2: Only 5 A's: The equation is: C(6,5) x 25
Case 3: All letters are A's: The Equation is: C(6,6)
Taking all of this into account, the real answer must be larger than Grogg's Answer.
The solution to Ms Q's problem
c) This question could be solved two ways. We could use rigorous casework. Or, we could use complementary counting. However, I will answer this question using casework.
In the previous, section, we highlighted the cases required to get the total answer.
Case 1: Only 4 A's. The equation is C(6,4) x (25 +25 x 24 x 2!).
Case 2: Only 5 A's: The equation is: C(6,5) x 25
Case 3: All letters are A's: The Equation is: C(6,6).
Now, we have to solve all of these cases.
Case 1: 18375
Case 2: 150
Case 3: 1.
Now that we have all of our cases, we can add them up, and our sum will be the answer.
18375 + 150 + 1 = 18526.
Therefore, there are 18526 ways to make license plates!!!
Hope that helps!!! :)
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