Playing the horses: In horse racing, one can make a trifecta bet by specifying 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 specifying which three horses will come in first, second, and third, without specifying the order Part 1 of 2 (a) In a ten-horse field, how many different ways can one make a trifecta bet? There are $\square$ ways to make a trifecta bet. Part 2 of 2 (b) In a ten-horse field, how many different ways can one make a box trifecta bet? There are $\square$ ways to make a box trifecta bet.

Step 1 ：In a ten-horse field, we are asked to find how many different ways one can make a trifecta bet and a box trifecta bet.

Step 2 ：For the trifecta bet, we need to find the number of ways to arrange 3 horses out of 10 in a specific order. This is a permutation problem, because the order of the horses matters. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.

Step 3 ：Substituting n = 10 and r = 3 into the formula, we get 10P3 = 10! / (10-3)! = 720. So, there are \(\boxed{720}\) ways to make a trifecta bet.

Step 4 ：For the box trifecta bet, we need to find the number of ways to choose 3 horses out of 10, without regard to order. This is a combination problem, because the order of the horses does not matter. The formula for combinations is nCr = n! / [r!(n-r)!].

Step 5 ：Substituting n = 10 and r = 3 into the formula, we get 10C3 = 10! / [3!(10-3)!] = 120. So, there are \(\boxed{120}\) ways to make a box trifecta bet.