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## couples together

Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

Is the answer 2/5 or 3/5 ? Official Answer is 2/5 but i think it should be 3/5. Can you help?

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24 ways both couples are together...
48 ways only one couple is together...
120 ways of arrangement ...
48 ways none of the couple is together

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Both couples sit together=2*2*!3=24
Assume A is couple 1 (together), B1 and B2 are couple 2 and C is single person

AB1CB2
AB2CB1
B1AB2C
B1ACB2
B2AB1C
B2ACB1
B1CAB2
CB1AB2
B2CAB1
CB2AB1
B2CB1A
B1CB2A

Only 1st couple sits together=2*12=24 (multiplying by 2 as couple A can also be arranged in 2 ways)
Only 2nd couple sits together=2*12=24 (same as above)

Total = 24+24+24=72

Total ways in which 5 persons can be seated = !5=120

Prob = 1-72/120=48/120=2/5

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