Marco83 wrote:
Fifteen runners from four different countries are competing in a tournament. Each country holds a qualifying heat to determine who its fastest runner is. These four runners then run a final race for first, second, and third place. If no country has more than one more runner than any other country, how many arrangements of prize winners are there?
A) 24
B) 384
C) 455
D) 1248
E) 2730
My approach is the following, but it doesn't lead to any of the answers:
First, we know that 3 countries have 4 runners, and 1 country has 3. This tells us that there are 4*4*4*3=192 possible final race rosters. Then we can ask ourselves: in how many ways can we arrange 4 runners? 4!=24, hence there are 192*24=4608 possible prize winner arrangements.
Where am I making a mistake?
hi Marco83,
nice problem...
i too m getting an answer that is not provided in the answer stem..
same as your method, but i think we have to multiply the answer by another 4, my answer is 18432.
total 15 members, so 4 from each of 3 countries and 3 from the fourth country... but the country from which 3 participants are competing can be any of the 4 countries...
so selecting 1 winner from each country can be done in, (4*4*4*3)*4 ways= 768 ways
now selecting and arranging (G,S and B) prize winners from the 4 participants can be done in (4C3)(3!) ways=24 ways
hence number of ways of selecting and arranging prize winners= 768*24=18432.
i m not sure about my answer... i hope some senior members of the club can help us...