Simple question relating to probability

In summary, the problem is asking for the number of arrangements of "TOYBOAT" without the T's being next to each other. The first step is to find the total number of arrangements without the restriction, which is 7!/(2!*2!) due to the repeated letters of T and B. To ensure that the T's are not next to each other, they can be treated as one letter, resulting in 7 possible positions for the "T" letter. This would give a total of 7 x 5! arrangements, taking into account the repeated letter "O."
  • #1
kevinf
90
0
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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  • #2
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:
 
  • #3
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
 
  • #4
yup … treat the two Ts as one letter :wink:
 
  • #5
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.
 
  • #6
kevinf said:
So then it would be 7 possible places that the t could sit in then.

uhh? :confused:

think again! :smile:
 
  • #7
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
 
  • #8
TT O Y B O A … only 6 letters! :wink:
 

1. What is probability?

Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, with 0 meaning the event is impossible and 1 meaning the event is certain to occur.

2. How is probability calculated?

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a 3 on a standard six-sided die is 1/6, as there is only one favorable outcome out of six possible outcomes.

3. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual results from experiments or observations and can vary from the theoretical probability.

4. How does sample size affect probability?

In general, as the sample size increases, the probability of an event occurring will approach the theoretical probability. This is known as the law of large numbers.

5. What are some real-life applications of probability?

Probability is used in a variety of fields, such as finance, insurance, and gambling. It can also be used to make predictions and decisions in areas such as weather forecasting, sports, and medical research. Additionally, understanding probability is important for critical thinking and decision making in everyday life.

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