Find the probability of forming a 2 digit number by using staring four prime number such that it is divisible by 3.

Options

A. `(1)/(3)`
B. `(1)/(4)`
C. `(5)/(16)`
D. `(1)/(2)`

Answer

To solve the problem of finding the probability of forming a 2-digit number using the first four prime numbers (2, 3, 5, and 7) such that the number is divisible by 3, we can follow these steps:

### Step 1: Identify the Prime Numbers
The first four prime numbers are:
- 2
- 3
- 5
- 7

### Step 2: Form All Possible 2-Digit Combinations
We can create 2-digit numbers by taking each prime number as the tens place and combining it with each of the four prime numbers in the units place. The combinations are:
- Starting with 2: 22, 23, 25, 27
- Starting with 3: 32, 33, 35, 37
- Starting with 5: 52, 53, 55, 57
- Starting with 7: 72, 73, 75, 77

This gives us a total of 16 combinations:
- 22, 23, 25, 27
- 32, 33, 35, 37
- 52, 53, 55, 57
- 72, 73, 75, 77

### Step 3: Count the Total Outcomes
The total number of 2-digit combinations formed is 16.

### Step 4: Identify Which Combinations are Divisible by 3
To check which of these numbers are divisible by 3, we can use the divisibility rule for 3 (a number is divisible by 3 if the sum of its digits is divisible by 3). The numbers that are divisible by 3 from our combinations are:
- 27 (2 + 7 = 9)
- 33 (3 + 3 = 6)
- 57 (5 + 7 = 12)
- 72 (7 + 2 = 9)
- 75 (7 + 5 = 12)

This gives us 5 favorable outcomes: 27, 33, 57, 72, and 75.

### Step 5: Calculate the Probability
The probability \( P \) of forming a 2-digit number that is divisible by 3 is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{16} \]

### Conclusion
Thus, the probability of forming a 2-digit number using the first four prime numbers that is divisible by 3 is \( \frac{5}{16} \).

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