There will be 8 people, arrange them in a row in a way that 2 must need to seat side by side, so the answer is $7!*2!$ but I get the same answer while using the following formula $ {\frac{8!}{2!*2!}} $. I cannot find a clear explanation about how the formula work.
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The explanation of $7!\cdot 2!$ is simple. You can arrange the people in two steps.
- First, take the two that must seat side by side as one person. You now have $7$ people and $7!$ possible ways to do this.
- Once you picked the arranging od $7$ people, you have to replace the "pair" person with two people. You have $2!$ ways to do this.
The explanation of $\frac{8!}{2!2!}$ is even simpler. Take an example when you would have $9$ people instead of $8$. Then you would have $8!\cdot 2!$ ways to do this, but $$8!2!\neq \frac{9!}{2!2!}.$$ Basically, the fact that $\frac{8!}{2!2!}$ is the same as $7!2!$ is merely a coincidence.
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I think you are right as ${\frac{8!}{2!2!}}=7!2$ it just dividing the 87654321 in 4 and getting 27654321 – gbox May 20 '14 at 07:36