5-digit numbers are to be formed using 2, 3, 5, 7, 9 without repeating the digits. If $$p$$ be the number of such numbers that exceed 20000 and $$q$$ be the number of those that lie between 30000 and 90000, then $$p : q$$ is:

We are forming 5-digit numbers using the digits 2, 3, 5, 7, 9 without repeating any digit. The total number of such numbers is the number of permutations of 5 distinct digits, which is $$5! = 120$$.

First, we find $$p$$, the number of numbers exceeding 20000. Since the smallest digit available is 2, the smallest possible 5-digit number we can form is 23579. This number is greater than 20000. All other numbers formed will also be greater than 20000 because they start with a digit at least 2 and have no leading zeros. Therefore, every number formed exceeds 20000. Hence, $$p = 5! = 120$$.

Next, we find $$q$$, the number of numbers lying between 30000 and 90000. This means the number must be greater than 30000 and less than 90000. We analyze this condition by considering the first digit:

• If the first digit is 2, the number will be in the range 20000 to 29999, which is less than 30000. So, numbers starting with 2 do not satisfy the condition.
• If the first digit is 9, the smallest number is 92357, which is greater than 90000. So, numbers starting with 9 exceed 90000 and do not satisfy the condition.
• If the first digit is 3, 5, or 7, the number will be between 30000 and 89999. For example:
  • Smallest number starting with 3: 32579 > 30000
  • Largest number starting with 3: 39752 < 90000
  • Smallest number starting with 5: 52379 > 30000
  • Largest number starting with 5: 59732 < 90000
  • Smallest number starting with 7: 72359 > 30000
  • Largest number starting with 7: 79532 < 90000
  Thus, numbers starting with 3, 5, or 7 satisfy the condition.

There are 3 choices for the first digit (3, 5, or 7). For each choice, the remaining 4 digits can be arranged in the next 4 positions in $$4! = 24$$ ways. Therefore, $$q = 3 \times 24 = 72$$.

Now, we have $$p = 120$$ and $$q = 72$$. The ratio $$p : q = 120 : 72$$. Simplifying this ratio by dividing both terms by their greatest common divisor, 24:

$$120 \div 24 = 5$$

$$72 \div 24 = 3$$

So, $$p : q = 5 : 3$$.

Comparing with the options:

• A. 6 : 5
• B. 3 : 2
• C. 4 : 3
• D. 5 : 3

The ratio 5:3 corresponds to option D.

Hence, the correct answer is Option D.