In how many ways can 5 men and their 5 women be arranged on a bench such that (a) each men sits next to his wife (b)one pair insists on sitting next to each other.

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First in case (a) you should look at each pair as one block since they must sit next to each other.  You can arrange 5 pairs in 5!=120 ways. Then you can arrange man and woman in 2 ways (mw or wm) for each of 5 pairs. 

Your result is `5! cdot 2^5=3840`

In case (b) you have 8 people and 1 pair which you can arrange in 9! ways and you can arrange man and woman in the pair in 2 ways (mw or wm).

Your result is `9! cdot 2=725760`

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