An army fleet is made with 4 cargo ships, 6 cruisers, 3 battleships, 7 destroyers and 3 aircraft carriers. The army has 7 cargo ships, 7 cruisers, 8 battleships, 10 destroyers and 5 aircraft carriers. How many different fleets can be made?
The way I did it was to split up the possible number of groups that can be formed from each ship class and then multiply the # of groups for each together.
n1 = # of groups of 4 cargo ships = 7C4 = 35
n2 = # of groups of 6 cruisers = 7C6 = 7
n3 = # of groups of 3 battleships = 8C3 = 56
n4 = # of groups of 7 destroyers = 10C7 = 120
n5 = # of groups of 3 aircraft carriers = 5C3 = 10
Total # of possible fleets = n1 x n2 x n3 x n4 x n5 = 16464000
I was wondering if this is correct, because to me this number seems very high?
Your answer is correct provided that each of the cargo ships, cruisers, battleships, destroyers and aircraft carriers is unique in nature. The answer may seem high at first but then realize that just swapping out one particular unit with another leads to a different combination. You often end up with huge numbers in combinatorics.