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June 15th, 2018, 09:41 PM  #1 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  How do you do this TATTOO problem?
(ex) How many words can you make using TATTOO if you have to begin with two T's? I try to solve it using combinations. Consider T T_ _ _ _ The first two positions are saved for the two T's. Now, you can arrange the remaining letters in 4!=24 ways. Then, I divide 24 by 2 because there are two O's. So, my answer is 12. But, the answer key says 9. How do you solve the problem? Thanks for your help. 
June 15th, 2018, 09:53 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 
So, we are essentially looking for the permutations of 4 objects, where 2 are identical. I would say: $\displaystyle N=\frac{4!}{2}=12$ I would do it the same way you did. 
June 15th, 2018, 10:18 PM  #3 
Newbie Joined: Jun 2018 From: USA Posts: 2 Thanks: 0 
The word should start with two Ts. Hence the third letter shall be either A or O but not T. Case 1: The third letter be A. Then the remaining 3 can be filled by 3!/2 =3. Case 2: The third letter be O. Then the remaining 3 can be filled by 3!=6. So the total number of words that can be formed is 6+3=9. 
June 15th, 2018, 11:47 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,549 Thanks: 1399 
If the 3rd letter can be a "T" then the answer is 12. If the 3rd letter cannot be a "T" then the answer is 9. It depends on your interpretation of "you have to begin with two T's" I'd say the answer is 12. The arrangement TTTAOO for example does start with 2 T's imo. 
June 16th, 2018, 02:44 PM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
Easier if TAOO = 1233

June 29th, 2018, 09:04 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
It's really a question of interpretation. Does "starts with two 't's" mean "starts with exactly two 't's" or "starts with at least two 't's".


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