akhp77 wrote:In a business school case competition, the top three teams receive cash prizes of $5,000, $3,000, and $2,000 respectively, while the remaining teams are not ranked and do not receive and prizes. There are 6 participating teams, names Team A, Team B, Team C, Team D, Team E, and Team F. If team A wins one of the prizes, Team B will also win one of the prizes. How many outcomes of the competition are possible?
There are 3 prize locations to be filled from 6 contestants.
The given constraint is unidirectional, "if A then B" is not logically same as "if B then A". Hence B could win a prize money without finding A collecting another.
If A wins a prize, two prizes are booked, and the remaining single prize could go to one of the remaining 4 contestants. Never forget that the three prizes could be swapped among the chosen three in 3! manners again. This could total to 1 × 1 × 4 × 3! = 24 ways.
These are the appearances of B when A was there too. Now, let's analyse the case when A didn't win a prize; now, B could or couldn't win a prize; also remember, there's no A now, so even if B is there among the prizes, this is not repeating a case with A there among the prizes.
If A doesn't win a prize, the 3 prize locations could now be filled from 5 contestants in 5P3 = 60 ways.
[spoiler]
24 + 60 = 84[/spoiler] many outcomes of the competition are possible.
