# How many students to award a prize? Combinations

Question: 80 tickets were sold to 50 engineering and 30 science students, one ticket per student. The tickets are entered in a prize draw. Five prizes are drawn: the Grand Prize, the Second Prize, and three more identical prizes. How many ways are there to award the prizes if the Grand Prize goes to an engineering student, and the Second Prize goes to a science student?

Attempt at a solution: I know that there are 3 prizes left after allocating the Grand Prize and the Second Prize to the engineering student, which is (78 choose 3). What I'm stuck on is how to allocate the Grand Prize and the Second Prize to the equation. Since the engineering student won the grand prize, there is 49 engineers who could win. Since the science student won the Second Prize, there are 29 scientists who could win a prize. Where do I go from here

• "Since the engineering student won the grand prize, there is 49 engineers who could win. Since the science student won the Second Prize, there are 29 scientists who could win a prize." - How/why do you fix this? I'd suggest do the counting from the beginning: How many students can win the grand prize? How many for the second prize? Now how do you distribute the remaining 3 prizes among 78 students (you've figured this part already)? – Sudarsan Mar 31 '15 at 21:55
• You can't start by $\binom{78}{3}$ since you don't know who won G and S... How about picking the 2 winners first, ie 50 choose 1, etc... and the remaining prizes after... – Theo Mar 31 '15 at 21:56