Questions: Six horses are entered in a race. If four horses are tied for first place, and there are no ties among the other two horses, in how many ways can the six horses cross the finish line?

Solution Steps

To solve this problem, we need to consider the different arrangements of the horses. First, we choose 4 horses out of 6 to tie for first place. Then, we arrange the remaining 2 horses in the second and third positions. The number of ways to choose 4 horses from 6 is given by the combination formula. The remaining 2 horses can be arranged in 2! (factorial) ways.

We need to select 4 horses out of the 6 to tie for first place. The number of ways to choose 4 horses from 6 is given by the combination formula:

\[ \binom{6}{4} = 15 \]

The remaining 2 horses can be arranged in the second and third positions. The number of ways to arrange 2 horses is given by the factorial of 2:

\[ 2! = 2 \]

To find the total number of ways the horses can cross the finish line, we multiply the number of ways to choose the 4 horses by the number of arrangements of the remaining 2 horses:

\[ \text{Total Ways} = \binom{6}{4} \times 2! = 15 \times 2 = 30 \]

Final Answer