Probability Question

Question

If 10 people consisting of 5 married couples, are randomly arranged in a row, find the probability that no person is next to their spouse. Give the numerical value of this probability.

Answer 1

We use the inclusion/exclusion principle

the number of possible orders is 10! = 3628800. To be consistent with the below, we can write

0 pairs = 2^0*PERMUT(5,0)*COMBIN(10,0)*FACT(10) = 3628800

1 pair = 2^1*PERMUT(5,1)*COMBIN(9,1)*FACT(8) = 3628800

2 pairs = 2^2*PERMUT(5,2)*COMBIN(8,2)*FACT(6) = 1612800

3 pairs = 2^3*PERMUT(5,3)*COMBIN(7,3)*FACT(4) = 403200

4 pairs = 2^4*PERMUT(5,4)*COMBIN(6,4)*FACT(2) = 57600

5 pairs = 2^5*PERMUT(5,5)*COMBIN(5,5)*FACT(0) = 3840

Then, 3628800 - 3628800 + 1612800 - 403200 + 57600 - 3840 = 1263360

P(0 couples next to each other) = 1263360/3628800 = 47/135 = .3481 (incidentally, as a repeating decimal, the 481 repeats).