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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 and  No two license plates are the same. How many possible Aopslandian license plates are there which contain exactly four 's, or exactly two 's, or both?"

Grogg got the answer $\binom{6}{4} \cdot 26^2 + \binom{6}{2} \cdot 26^4,$ 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.

 Jul 7, 2023
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(a) Grogg's answer was probably arrived at by considering the two cases of exactly 4 A's and exactly 2 A's separately. For exactly 4 A's, there are (46​) ways to choose the 4 A's, and then 26 choices for each of the remaining two letters. For exactly 2 A's, there are (26​) ways to choose the 2 A's, and then 26 choices for each of the remaining four letters. So Grogg's answer is (46​)⋅262+(26​)⋅264.

 

(b) Grogg's answer is wrong because it does not account for license plates with 5 A's or 6 A's. There are (56​) ways to choose the 5 A's, and then 1 choice for the remaining letter, so there are (56​)=6 license plates with 5 A's. There is only 1 license plate with 6 A's. So Grogg's answer should be (46​)⋅262+(26​)⋅264+6+1. The correct answer is larger than Grogg's answer because Grogg's answer does not account for all the possible license plates.

 

(c) The correct answer to Ms. Q's problem is (46​)⋅262+(26​)⋅264+6+1=15⋅676+1560+7=26353. This is the number of license plates that contain exactly 4 A's, exactly 2 A's, 5 A's, or 6 A's.

Here is a more detailed explanation of how to solve the problem:

Case 1: Exactly 4 A's. There are (46​) ways to choose the 4 A's, and then 26 choices for each of the remaining two letters. So there are (46​)⋅262=15⋅676=10140 license plates with exactly 4 A's.

Case 2: Exactly 2 A's. There are (26​) ways to choose the 2 A's, and then 26 choices for each of the remaining four letters. So there are (26​)⋅264=1560 license plates with exactly 2 A's.

Case 3: 5 A's. There are (56​) ways to choose the 5 A's, and then 1 choice for the remaining letter. So there is 1 license plate with 5 A's.

Case 4: 6 A's. There is only 1 license plate with 6 A's.

The total number of license plates is 10140+1560+1+1=26353​.

 Jul 7, 2023

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