seven horses are entered in a race. If two horses are tied for first​ place, and there are no ties among the other five ​horses, in how many ways can the seven horses cross the finish​ line?

Old MacDonald had some horses, cows, and pigs, on his farm. He had 69 water troughs from which these animals drink. Horses drank only from the horse troughs, exactly two horses for each trough. Cows drank from cow troughs, exactly three cows per trough and pigs drank from pig troughs, exactly eight pigs per trough. Old MacDonald’s farm has the same number of horses, cows, and pigs. How many animals does Old MacDonald have on his farm?

To determine the number of ways the seven horses can cross the finish line, we need to consider the possible scenarios for the tied horses.

Since two horses are tied for first place, there are two scenarios:

Scenario 1: The tied horses finish in first and second position, followed by the remaining five horses. This can be arranged in 2! ways (2 factorial).

Scenario 2: The tied horses finish in second and third position, followed by the remaining five horses. This can also be arranged in 2! ways.

For the remaining five horses (excluding the tied horses), they can finish in the remaining positions in 5! ways (5 factorial).

Therefore, the total number of ways the seven horses can finish the race is:

Total = (Scenario 1) + (Scenario 2) = 2! * 5! + 2! * 5!

= 2 * 120 + 2 * 120

= 240 + 240

= 480

Therefore, there are 480 possible ways the seven horses can cross the finish line.

To solve this problem, we can consider the cases separately.

Case 1: Two horses tie for first place.

Since two horses are tied for first place, we can arrange these two horses in two ways: 2! = 2. The remaining five horses can then be arranged in the remaining positions in 5! = 120 ways. Therefore, for this case, the total number of arrangements is 2 * 120 = 240.

Case 2: No ties among the other five horses.

For the remaining five horses, there are no ties, so we can arrange them in 5! = 120 ways.

Total number of arrangements = (Number of arrangements in Case 1) * (Number of arrangements in Case 2) = 240 * 120 = 28,800.

Therefore, there are 28,800 ways for the seven horses to cross the finish line.

So first you want to know how many pairs I can form from the 7 horses , then arranging the other 5

That would be C(7,2) or 21 for the first part
- the remaining 5 horses can be arranged in
5! or 120 ways

So there are 21(120) or 2520 ways to finish the race