There is a group of 5 people among which there is one couple. In how m

There is a group of 5 people among which there is one couple. In how many ways can these 5 people be seated in a row having 5 chairs if the couple is to be seated next to each other?

120

Answer is Right!

Answer is Wrong!

This question was previously asked in

UPSC CAPF – 2022

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To solve this problem, we treat the couple as a single unit. This means we are arranging 4 entities: the couple (as one unit) and the remaining 3 individuals. These 4 entities can be arranged in 4! ways. Since the two members of the couple can swap positions within their unit, there are 2! ways for them to be seated relative to each other. The total number of ways is the product of the ways to arrange the units and the ways to arrange within the unit: 4! × 2! = 24 × 2 = 48.

– Treat the constrained group (the couple) as a single unit.
– Calculate the number of permutations of the resulting units (the couple unit + the other individuals).
– Calculate the number of permutations within the constrained group (the couple).
– The total number of arrangements is the product of these two numbers.

If there were no constraints, 5 people could be seated in 5! = 120 ways. The constraint that the couple sits together reduces the number of possibilities significantly. This method of treating a group that must stay together as a single unit is a standard technique in permutation problems.

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