National Radio Broadcast will put a contest to guess five winners out of twelve local boxers who will compete to win the best 5 boxers. All twelve boxers are equally good so the chance of winning is based on luck. The order of the winners is not important. The contestant who guess all five winners correctly will get a $ 35,000.00 prize.
(i) How many contestants will be required to participate in the guessing contest if every contestant has different combination of winners in his/her guess to ensure that there will be exactly one contestant who guess all five winners?
(ii) Suppose that all five winners have been announced and the contestant who guesses all five winners correctly did not show up to claim his/her prize money. So the jury decided to share the prize money equally among contestants who guess four winners correctly. How much money did each contestant who guessed four correct winners get?
In the first case we need all the combinations to be played and so: $$\binom{12}{5} = \dfrac{12!}{5!7!}=792 $$
That means we need at least $792$ players in order to place all the possible bets.
In the second question, supposing that all 792 contestants placed their bets, we have five winners from which we may take four and also from the $8$ remainder ($12-4$ winners) we must take one more to place the bet.
That leave us with: $$\binom{5}{4} \cdot \binom{8}{1} = 5 \cdot 8 = 40$$
$40$ contestants that guessed the four correct winners, including the guy that guessed all the five correct.
Because of this, we had $39$ contestants that should divide the prize and each one should receive: $$\dfrac{35000}{39} \approx 897.44$$
If we consider that by the jury rules the guy that guessed the five should also be allowed to take the prize, each contestant should earn $875,00