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.

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?

Step 1of 3:

In a horse racing, one can make trifecta 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. We have to find

(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 trifecta box bet.

Step 2of 3:

Here in an eight-horse field one has to select the trifecta by specifying which three horses will come first.So if there is N horses. If a person select one horse by claiming that it will come in the first position. Then he has (N-1) different choices to select a horse by claiming that it will come in the second position. Similarly he has (N-2) different choices to select a horse by claiming that it will come in the third position.

Since all this selection are independently the total possible ways to make a trifecta bet

= N (N-1)

So in the case of eight horses one can make a trifecta = 876

= 336. Ways

That means in an eight horse field one can make a trifecta bet in 336 ways.