Two numbers a and b are chosen simultaneously from the set of integers 1, 2, 3, ….., 39, then the probability that the equation `7a-9b=0` is satisfied is

Options

A. `(1)/(247)`
B. `(2)/(247)`
C. `(4)/(741)`
D. `(5)/(741)`

Answer

To find the probability that the equation \(7a - 9b = 0\) is satisfied when two numbers \(a\) and \(b\) are chosen from the set of integers \(1, 2, 3, \ldots, 39\), we can follow these steps:

### Step 1: Understand the Equation
The equation \(7a - 9b = 0\) can be rearranged to find a relationship between \(a\) and \(b\):
\[ 7a = 9b \implies \frac{a}{b} = \frac{9}{7} \implies a = \frac{9}{7}b \]
This means \(a\) must be a multiple of \(9\) and \(b\) must be a multiple of \(7\).

### Step 2: Find Possible Values for \(b\)
Since \(b\) must be an integer from \(1\) to \(39\) and a multiple of \(7\), the possible values for \(b\) are:
\[ 7, 14, 21, 28, 35 \]
This gives us \(5\) possible values for \(b\).

### Step 3: Calculate Corresponding Values for \(a\)
For each value of \(b\), we can calculate the corresponding value of \(a\):
- If \(b = 7\), then \(a = \frac{9}{7} \times 7 = 9\)
- If \(b = 14\), then \(a = \frac{9}{7} \times 14 = 18\)
- If \(b = 21\), then \(a = \frac{9}{7} \times 21 = 27\)
- If \(b = 28\), then \(a = \frac{9}{7} \times 28 = 36\)
- If \(b = 35\), then \(a = \frac{9}{7} \times 35 = 45\) (not valid since \(a\) must be ≤ 39)

Thus, the valid pairs \((a, b)\) that satisfy the equation are:
1. \((9, 7)\)
2. \((18, 14)\)
3. \((27, 21)\)
4. \((36, 28)\)

This gives us a total of \(4\) favorable cases.

### Step 4: Calculate Total Cases
The total number of ways to choose \(2\) numbers from \(39\) is given by the combination formula:
\[ \text{Total cases} = \binom{39}{2} = \frac{39 \times 38}{2} = 741 \]

### Step 5: Calculate Probability
The probability \(P\) that the equation \(7a - 9b = 0\) is satisfied is given by:
\[ P = \frac{\text{Number of favorable cases}}{\text{Total cases}} = \frac{4}{741} \]

### Final Answer
Thus, the probability that the equation \(7a - 9b = 0\) is satisfied is:
\[ \boxed{\frac{4}{741}} \]