The number of ways in which first, second and third prizes can be given to 5 competitors is

Options

A. 10
B. 60
C. 15
D. 125

Answer

To find the number of ways in which first, second, and third prizes can be given to 5 competitors, we can follow these steps:

### Step 1: Choose 3 Competitors
First, we need to choose 3 competitors out of the 5 to receive the prizes. The number of ways to choose 3 competitors from 5 is given by the combination formula:

\[ \text{Number of ways to choose 3 out of 5} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} \]

### Step 2: Calculate \(\binom{5}{3}\)
Calculating \(\binom{5}{3}\):

\[ \binom{5}{3} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Assign Prizes
Now that we have chosen 3 competitors, we need to assign the first, second, and third prizes to these 3 competitors. The number of ways to assign 3 distinct prizes to 3 competitors is given by the factorial of the number of competitors:

\[ \text{Number of ways to assign prizes} = 3! = 6 \]

### Step 4: Total Number of Ways
Now, we can find the total number of ways to award the prizes by multiplying the number of ways to choose the competitors by the number of ways to assign the prizes:

\[ \text{Total ways} = \binom{5}{3} \times 3! = 10 \times 6 = 60 \]

Thus, the total number of ways in which the first, second, and third prizes can be given to 5 competitors is **60**.