In horse racing, one can make a trifecta bet by speci- fying which horse will come in first, which will come in second, and which will come in third, in the correct order. one can make a box trifecta bet by specify- ing which three horses will come in first, second, and third, without specifying the order. a. in an eight-horse field, how many different ways can one make a trifecta bet? b. in an eight-horse field, how many different ways can one make a box trifecta bet?

a) 336

b)56

Step-by-step explanation:

The difference is if the order matters or not.

If the order matters, you should use the permutation mathematical function, while if the order doesn't matter, you should use combinations.

a)

For the first place you have 8 posible horses, then after any horse wins, you still have 7 compiting for second place. similarly, for the third place you have 6 horses. To know the total amount of posble permutations, just multiply these numbers

mathematically the formula would be:

where n is the total posible objects and k is the amount of objects you pick

b)

If order doesn't matter, you need to find all the possible permutations and then eliminate the groups that are redundant. If your horses are named A, B, C, D, E, F, G and H, you need to pick 3 of them, for example C, F,G. In combinations, CFG is the same as FGC and GCF, therefore you need to eliminate these options. to do so just divide the permutations value by all the possible groups you just picked, which can be calcuted using factorial: