in a minibus there are 16 seats for passengers. How many possible seating arrangements are there for 5 passengers?

the answer in the book is 210. no idea how you get that. isn't it a simple $\displaystyle ^{16}P_5$?

furthermore in a similar question: there are 10 seats in the front row at a theatre, 6 people are shwon to this row. In how many different ways can they be seated if there are no restrictions? using $\displaystyle ^{10}P_6$ gives the textbook answer.

and there is one more question, also similar. rory has to place colored pegs into holes in a board. he has 6 identical red pegs and the board has 10 holes. how many different arrangements are there for placing the 6 pegs and leaving 4 empty holes? using $\displaystyle ^{10}P_6$ will give the textbook answer. however, these are identical pegs, not distinct people as in the previous 2 questions. using $\displaystyle ^{10}P_6$ means that arrangements such as {red peg 1, then red peg 2}, and {red peg 2, then red peg 1} are counted as 2 different arrangements. shouldn't there be more calculations involved, such as $\displaystyle ^{10}P_6 \div 6!$?

Your answers are correct for the problems as written. I wonder if the answer of 210 may be for a completely different problem - one whose answer is permutation P(7,3).

May I ask that why is it P(7,3)? I don't quite understand the questions of which the seats are more than the no. of ppl.

Originally Posted by deeyl

May I ask that why is it P(7,3)? I don't quite understand the questions of which the seats are more than the no. of ppl.

For the problem of 5 people and 16 seats, the number of possible seating arrangements is P(16,5) = 16 x 15 x 14 x 13 x 12, assuming that the order of people is important (i.e. if the first 5 setas are occupied a seating order of ABCDE is considered to be different than BACDE). If the question was worded a bit differently, such as "how many possible combinations of occupied seats are there?" then the answer is C(16,5) = 16 x 15 x 14 x 13 x 12/5!. In either case the answer is not P(7,3) - I was simply noting that the answer of 210 given in the book is incorrect and happens to be equivalent to P(7,3).

Actually it turns out that 210 is also equivalent to C(10,4). So perhaps the answer in the book may be for a problem involving how many ways seats can be occupied if there are 10 seats and 4 passengers.