Six horses take part in a race. In how many ways can these horses come in the first, second and third place, if a particular horse is among the three winners (Assume NO Ties)?

Answer

To solve the problem of how many ways six horses can come in the first, second, and third places, given that a particular horse is among the three winners, we can follow these steps:

### Step 1: Identify the Total Horses and the Particular Horse
We have a total of 6 horses, and we know that one specific horse (let's call it Horse A) must finish in one of the top three positions.

### Step 2: Determine the Position of the Particular Horse
Since Horse A must be in one of the top three positions (1st, 2nd, or 3rd), we have 3 choices for where to place Horse A.

### Step 3: Choose the Remaining Horses
After placing Horse A, we need to fill the remaining two positions with the other horses. Since Horse A is already placed, we have 5 horses left to choose from.

### Step 4: Select and Arrange the Remaining Horses
We need to select 2 horses from the remaining 5 horses and arrange them in the remaining two positions. The number of ways to choose 2 horses from 5 is given by the combination formula \( C(n, r) \), where \( n \) is the total number of horses left, and \( r \) is the number of horses to choose.

However, since the order matters (1st and 2nd place are different), we will use permutations. The number of ways to arrange 2 horses from 5 is given by \( P(5, 2) = 5 \times 4 \).

### Step 5: Calculate the Total Arrangements
Now, we multiply the number of choices for Horse A's position by the number of arrangements of the remaining horses:
- Choices for Horse A's position: 3
- Arrangements of the remaining horses: \( 5 \times 4 = 20 \)

Thus, the total number of ways is:
\[ \text{Total Ways} = 3 \times (5 \times 4) = 3 \times 20 = 60 \]

### Final Answer
The total number of ways the horses can come in the first, second, and third places, with the condition that a particular horse is among the winners, is **60**.

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