How Many Ways to Arrange TOYBOAT With No Adjacent T's?

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Discussion Overview

The discussion revolves around the combinatorial problem of arranging the letters in "TOYBOAT" with the condition that the two T's cannot be adjacent. Participants explore different approaches to solve the problem, including calculating total arrangements and considering complementary counting.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant calculates the total arrangements of "TOYBOAT" without restrictions as 7!/(2!*2!), accounting for repeated letters.
  • Another suggests using complementary counting by first finding the arrangements where the T's are adjacent.
  • There is confusion about how to treat the T's as a single unit and the implications for counting arrangements.
  • Participants discuss the number of spots available for placing the T's after treating them as one letter, with differing opinions on whether there are 6 or 7 spots.
  • Concerns are raised about accounting for repeated letters, particularly the O's and A's, in the arrangements.

Areas of Agreement / Disagreement

Participants express differing views on the counting method and the number of available spots for the T's, indicating that there is no consensus on the approach to take.

Contextual Notes

Participants have not fully resolved how to handle the repeated letters in their calculations, and there is uncertainty about the correct number of arrangements based on the proposed methods.

kevinf
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Hi, i have a problem that asks how many arrangements of "TOYBOAT" are there if the T's can not be next to each other.
I know the first step is to find the total without the restriction, which is 7!/(2!*2!). the 2 2! represents the repeated letters of T and B but I'm not sure how to make it so that the T's can not be next to each other. I've listed the different ways that the T's could sit so that they are not next to each other, which is 30. any hints guys? it seems simple but for some reason i can't wrap my head around it
 
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hi kevinf! :smile:

it often helps to go for the opposite :wink: …

in this case, to find the number of ways in which the Ts are next to each other! :biggrin:
 
sorry but I'm not quite understsanding how i would get the answer that way but it would be 6 ways? and each of the 6 ways have 5! ways of arranging the other letters? or maybe not, since the O's are also repeated.

sorry lol can you elaborate on that a little more? sorry
 
yup … treat the two Ts as one letter :wink:
 
So then it would be 7 possible places that the t could sit in then. Then wouldn't it be 7 x 5! . But what about the repeated a.
 
kevinf said:
So then it would be 7 possible places that the t could sit in then.

uhh? :confused:

think again! :smile:​
 
Lol if I understood you correctly, after making t one letter wouldn't there be 7 spots where t could go instead of 6 because t is now one letter
 
TT O Y B O A … only 6 letters! :wink:
 

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