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

Set this up as a multiplication principle:

• How many ways can you choose the one engineering student who gets the grand prize?
• How many ways can you choose the one science student who gets the second prize?
• How many ways can you choose three students out of the remaining to get one of the remaining prizes (where order doesn't matter)?

- For the first step, there are 50 engineers, we want to choose one. - For the second step, there are 30 science students, we want to chose one. - For the third step, there are 78 remaining students, we want to choose three.

Multiplication principle says that the total number of ways of accomplishing a task is the product of the number of ways of accomplishing each step (assuming every outcome is counted at least once and every outcome is counted at most once).