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

Question

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 ?

Answer 1

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.

Answer 2

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 $\dfrac{8!\cdot 3!}{2!}$.

Answer 3

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 $\frac{8!×3!}{2!}$.