Question

10 horses from three different stables participate in a race. 5 horses
come from stable A, 3 from stable B and 2 from stable C.

A) Calculate the number of possible results for the ten horses.

B) Calculate the number of possible results by grouping the horses
by stable.

c) For a trifecta(tierced), we choose the horses that will win the three first places in order. Calculate the number of possible trifecta(tierced).

d) Calculate the number of possible trifecta(tierced) by grouping the horses by stable.

e)For a trio, we choose the horses that will win the three first places. Calculate the number of possible trios.

theres is 5 horses In stable A,

3 horses in stable B

2 Horses in stable C

Answer 1

A. Using the multiplication principle of counting,

The first horse can come at any one of the 10 places in the race, the second horse can come at any one of the 9 remaining places, the third horse can have any one of the remaining 8 places and so on until the last horse will have 1 position to come at.

Thus the total number of possible results is 10*9*8*7*6*5*4*3*2*1 = 10! = 3628800

B. Using the same approach as above, the 5 horses of stable A can have 5! possible results, the 3 horses of stable B can have 3! possible results and the 2 horses of stable C can have 2! possible results. Also, the three stables can have 3! possbile results among themselves. Hence, total number of possible results are 5! * 3! * 2! * 3! = 8640

C. The first place can be taken by any one of the 10 horses, the second by any of the remaining 9 and the third by any one of the remaining 8. Hence, the possible number of trifectas is 10*9*8 = 720

D. There are several scenarios possbile.

i. The first three are from stable A.Then 5*4*3 = 60 possibilities are there.

ii. The first three are from stable B. Then 3*2*1=6 possibilites are there.

iii. Since there are only 2 horses in stable C, one horse will be from either stable A or B. Thus there are two cases.

a. The third horse is from stable A. In this case, the third horse can be chosen in 5 ways. Now the three horses can be placed in the top three positions in 3*2*1 ways. Thus the total number of ways is 5*3*2*1 = 30

b. The third horse is from stable B. In this case, the third horse can be chosen in 3 ways. Now the three horses can be placed in the top three positions in 3*2*1 ways. Thus the total number of ways is 3*3*2*1 = 18

Now, since each of the abover mentioned ways are mutually exclusive, we can add the possibilites. Hence the total number of ways is 60 + 6 + 30 +18 = 114

E. Since order doesn't matter, we just have to choose 3 horses out of 10. Thus there are = 120 ways.