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

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 ?

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

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• 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. – MisterRiemann Nov 24 '18 at 15:00
• Be careful. TECHNOLOGY has ten letters, so you have a block of three letters and seven other letters to arrange. – N. F. Taussig Nov 24 '18 at 15:04

## 3 Answers

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 $$\frac{8!×3!}{2!}$$.

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.

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

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