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Given TECHNOLOGY , find number of distinguishable ways the letters can be arranged in which letters T,E and N are together

This is my working-

$3! \cdot \frac{7!}{2!} $

is this correct ?

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closed as off-topic by Don Thousand, MisterRiemann, NCh, Shailesh, John B Dec 1 '18 at 0:45

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    $\begingroup$ Solutions to such exercises should usually be presented in such a way that one can understand how you reached the answer, i.e. you should explain how you got that particular number, instead of just presenting the final answer. $\endgroup$ – MisterRiemann Nov 24 '18 at 15:00
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    $\begingroup$ Be careful. TECHNOLOGY has ten letters, so you have a block of three letters and seven other letters to arrange. $\endgroup$ – N. F. Taussig Nov 24 '18 at 15:04
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Assume $T,E, N $ as single letter therefore, total number of letters in the word technology is 8 this can be arranged in $8!$ ways and number of ways in which $T,E, N$ can be arranged $3!$ ways and since $O$ is repeating two times hence answer is $\frac{8!×3!}{2!}$.

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I won't give the exact answer as then I'm unsure of the answer's helpfulness for other counting-type questions, but I hope that asking the following questions will lead you to the correct answer:

  • Can you explain your working for getting the 7!, 3! and 2! ?
  • Will using 7! include the possibilities where T, E, N are together but are located elsewhere?
  • In how many positions can the group of three letters be placed together?
  • Does using 7! account for all of these positions?

Perhaps these questions will lead you to the correct solution.

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Consider $\text{TEN}$ together as a block and all other letters as single block. Then you have $8$ blocks. So there are $8!$ permutations possible and $3!$ permutations of $TEN$. Also the letter $\text{O}$ is not distinguishable.

So total number of ways is $\dfrac{8!\cdot 3!}{2!}$.

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  • $\begingroup$ @N.F.Taussig: Sorry. My bad. $\endgroup$ – Yadati Kiran Nov 24 '18 at 16:07

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