Question 1203679
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Part (a)


If we could tell the "A"s apart, then there would be 5! = 5*4*3*2*1 = 120 different five-letter permutations. 


But because we cannot tell the "A"s apart, we must divide by 2 which corrects the erroneous double-count.


We'll have 120/2 = 60 different five-letter words possible.


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Part (b)


We have these four possible cases:<ol><li>The letter "A" is used twice</li><li>The letter "A" is used exactly once in the 1st slot.</li><li>The letter "A" is used exactly once in the 2nd slot.</li><li>The letter "A" is not chosen at all.</li></ol>Case 1 has exactly one outcome. That outcome is AA.


Case 2 has 3 choices for the 2nd slot, which means we have the 3 permutations AT, AR, AI


Case 3 also has 3 outcomes. The logic is similar to case 2.


Case 4 has 3*2 = 6 permutations of the letters {T,R,I}


Add up the results
1+3+3+6 = 13
There are 13 different two letter sequences



The entire list is<ol><li>AA</li><li>AT</li><li>AR</li><li>AI</li><li>TA</li><li>TR</li><li>TI</li><li>RA</li><li>RT</li><li>RI</li><li>IA</li><li>IT</li><li>IR</li></ol>A combinatorics calculator I recommend is this
<a href="https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html">https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html</a>
It computes the nPr value, and also lists the different permutations. 
Refer to the "List Them" section on that page.


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Part (c)


The previous part calculated 13 different two-letter words.


Let's calculate how many three-letter sequences are possible.


We have these cases<ol><li>The letter "A" is used twice.</li><li>The letter "A" is used exactly once.</li><li>The letter "A" is not chosen at all.</li></ol>


Case 1
There are 3 slots to pick from where the letters T, R, or I will go.
After that slot is chosen, the other two slots are locked in as "A"
Therefore, we have 3*3 = 9 ways to do case 1.


Case 2
There are 3 slots to pick for the letter "A".
We have two more slots and 3 more letters to pick from to fill said slots. We'll have 3*2 = 6 permutations of those other letters.
Overall we have 3*6 = 18 ways to carry out case 2.


Case 3
There are 3*2*1 = 6 permutations of the letters {T,R,I}


Add up the results
9+18+6 = 33
We find there are 33 different three-letter words.


I'll let you do the scratch work to determine how many four-letter words are possible. 
You should find there are 60 different four-letter words possible.


To summarize we have...<ul><li>60 ways to form a five-letter word.</li><li>60 ways to form a four-letter word.</li><li>33 ways to form a three-letter word.</li><li>13 ways to form a two-letter word.</li></ul>There are 60+60+33+13 = 166 different words possible when we can only select letters from the list {A,T,A,R,I} where "A" can be used at most twice, but the other letters at most once.


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Answers:
(a) <font color=red>60</font>
(b) <font color=red>13</font>
(c) <font color=red>166</font>
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