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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?

RishtarCode47
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  • Usually in those combinatorics problems the number you get as the result isn't so important as the reasoning you used, Maybe you can even let it expressed as a product and not calculating the final result (because it can be very large) – Alejandro Tolcachier Jan 15 '20 at 15:28
  • "This number seems very high" and the number of different orders in which cards from a standard deck of $52$ cards can appear is $52! \approx 8\times 10^{67}$. Answers in questions like this can very quickly grow quite large. – JMoravitz Jan 15 '20 at 15:36

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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.

Sam
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