Three different numbers are selected at random from the set `A = (1, 2, 3,...., 10).` The probability that the product of two of the numbers is equal to the third is

Options

A. `(3)/(4)`
B. `(1)/(40)`
C. `(1)/(8)`
D. `(39)/(40)`

Answer

To solve the problem of finding the probability that the product of two of the selected numbers is equal to the third number when three different numbers are chosen from the set \( A = \{1, 2, 3, \ldots, 10\} \), we can follow these steps:

### Step 1: Identify the total number of ways to choose 3 different numbers from the set.

The total number of ways to choose 3 different numbers from a set of 10 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Total outcomes} = \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 2: Identify the favorable outcomes where the product of two numbers equals the third.

We need to find combinations of three numbers \( (a, b, c) \) such that \( ab = c \). We can systematically check combinations:

1. **For \( c = 6 \)**:
- \( (2, 3, 6) \) since \( 2 \times 3 = 6 \)

2. **For \( c = 8 \)**:
- \( (2, 4, 8) \) since \( 2 \times 4 = 8 \)

3. **For \( c = 10 \)**:
- \( (2, 5, 10) \) since \( 2 \times 5 = 10 \)

These combinations yield 3 favorable outcomes: \( (2, 3, 6) \), \( (2, 4, 8) \), and \( (2, 5, 10) \).

### Step 3: Calculate the probability.

The probability \( P \) that the product of two of the selected numbers equals the third is given by the ratio of the number of favorable outcomes to the total outcomes.

\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{120} = \frac{1}{40} \]

### Final Answer:
The probability that the product of two of the numbers is equal to the third is \( \frac{1}{40} \).

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