Mr and Mrs Smith have invited 9 of their friends and their spouses for a party at the Waikiki Beach resort. They stand for a group photograph. If Mr Smith never stands next to Mrs Smith (as he says they are always together otherwise). How many ways the group can be arranged in a row for the photograph?
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Mr. and Mrs. Smith invited 9 of thier friends and their spouses.
∴, total 20 persons are present there.
They can be arranged in 20! ways.
Now, if we consider that Mr. Smith stands with Mrs. Smith always then we can consider them as 1 person. Then remining (20-1)=19 persons can be arranged in 19! ways and the position of Mr. and Mrs. Smith can be arranged in 2! ways.
∴, the required number of arrangement is:
20!-19!×2!
=20×19!-19!×2×1
=19!(20-2)
=18×19!
∴, total 20 persons are present there.
They can be arranged in 20! ways.
Now, if we consider that Mr. Smith stands with Mrs. Smith always then we can consider them as 1 person. Then remining (20-1)=19 persons can be arranged in 19! ways and the position of Mr. and Mrs. Smith can be arranged in 2! ways.
∴, the required number of arrangement is:
20!-19!×2!
=20×19!-19!×2×1
=19!(20-2)
=18×19!
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