There are 3 entities of ingredient A, 8 of ingredient B, 12 of C and 7 of D.
Person $P_3$ may consume his dish iif it contains an even number of ingredients. Person $P_4$ requires that his dish contains at least 2 entities of ingredient B and person $P_1$ wants that if his dish contains any number of A's, then it contains exactly as much C's. Persons $P_2$ and $P_5$ have no special preferences.
In how many different ways may one cook dishes for all? (a dish contains at least 1 ingredient)
This problem is giving me one hell of a time. I know I must utilise the inclusion exclusion principle but I am really stuck.
Any help on this is greatly appreciated.