A-Level Probability: worked solution

Question

Eight tourists, being 2 married couples and 4 single travellers, share two taxis X and Y to the airport, each holding 4 people. They split into two fours, one per taxi. (i) In how many ways if each couple must ride together in one taxi? Each taxi seats 1 in front and 3 in the back (see figure). (ii) How many seating arrangements have each married couple side by side in the back?

Name the people: couple 1 is (A1, A2), couple 2 is (B1, B2), and the four single travellers are S1, S2, S3, S4. The two taxis X and Y are different vehicles, each carrying exactly 4 people. (i) Here we only choose WHO is in each taxi, and neither couple may be split. Treat each couple as one block that must stay whole. There are two possibilities. Case A: both couples travel in the same taxi. That taxi is then already full (2 + 2 = 4 people), so all four singles fill the other taxi. The only choice left is which taxi holds the couples: 2 ways. Case B: the couples travel in different taxis. Choose the taxi for couple 1 (couple 2 must then take the other): 2 ways. Each taxi now has 2 seats left, and there are 4 singles to share out. Choose the 2 singles who join taxi X: 4C2 = 6 ways; the remaining 2 go to taxi Y: 2C2 = 1 way. So Case B gives 2 x 6 x 1 = 12 ways. Total = 2 + 12 = 14 ways. (ii) Now the seats matter: each taxi has 1 seat in front and 3 seats in a row at the back. 'Side by side in the back' means the husband and wife occupy two ADJACENT seats of that row of three. A row of 3 seats has only 2 adjacent pairs (left-middle and middle-right). Two couples could not both sit as adjacent pairs in the same back row, since that would need 4 back seats. So the two couples must be in different taxis. Step 1: decide which couple is in taxi X (the other takes taxi Y): 2 ways. Step 2: seat the couple inside its own taxi. Choose which adjacent pair of back seats they use: 2 ways. Decide which of the two sits on the left: 2 ways. So 2 x 2 = 4 arrangements for that couple, and the same 4 for the other couple: 4 x 4 = 16 ways. Step 3: seat the four singles. Each taxi now has exactly 2 empty seats (the front seat and the one remaining back seat), so there are 4 empty seats in total and 4 singles to place in them. That is an arrangement of 4 distinct people in 4 distinct seats: 4! = 24 ways. Total = 2 x 16 x 24 = 768 seating arrangements.