Book no 12, Combinatorics and Probability, question 79, page 128
Jake, Lena, Fred, John and Inna need to drive home from a corporate reception in an SUV that can seat 7 people. If only Inna or Jake can drive, how many seat allocations are possible?
My question is:
the solution states that "since the driver's seat is occupied, we have to allocate the remaining 4 people among the 6 passenger seats. Because the order of the seat assignments is important, we will use the permutations formula"
Considering the fact that the restriction placed on the question is that the driver's seat can only be occupied by either Inna or Jake, If we were to place Jake in the driver's seat, why would the order of selection matter when we are choosing where to place the remaining 4 passengers among 6 seats?
In other words, why do we need to use the permutations formula instead of the combinations formula?
The way I solved this problem is illustrated below -
Assuming Jake is the driver,
We have 4 remaining people that can be seated among 6 seats
Order of selection for the REMAINING people does not matter
Hence, combinations formula is used to find out number of ways 4 people can be seated in 6 places
That is, 6C4 = 6!/(4!x2!) = 15
Now since Inna could also be the driver, we have another set of seating arrangments:
therefore total number of seat allocations = 15x2 = 30
Answer choice A