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:

Options

A. A. 5:2
B. B. 3:2
C. C. 7:5
D. D. None of the above

Answer

To solve the problem step by step, we need to find the values of \( p \) and \( q \) based on the conditions given in the question.

### Step 1: Calculate \( p \) (Numbers greater than 20000)

1. **Identify the first digit**: The first digit must be either 2, 3, 5, 7, or 9. Since we are looking for numbers greater than 20000, any of these digits can be used.
2. **Count the total arrangements**: Since we can use any of the 5 digits for the first position, and the remaining 4 digits can be arranged in any order, we calculate:
\[ \text{Total arrangements} = 5! = 120 \]
So, \( p = 120 \).

### Step 2: Calculate \( q \) (Numbers between 30000 and 90000)

1. **Identify the valid first digits**: The first digit must be either 3, 5, 7, or 9 (it cannot be 2 as that would make the number less than 30000).
2. **Count the valid first digits**: The first digit can be 3, 5, 7, or 9, giving us 4 options.
3. **Count the arrangements for each case**:
- After choosing the first digit, we have 4 digits left to arrange in the remaining 4 positions.
- The number of arrangements for the remaining 4 digits is:
\[ 4! = 24 \]
4. **Total arrangements for \( q \)**: Since we have 4 choices for the first digit and 24 arrangements for the remaining digits, we calculate:
\[ q = 4 \times 24 = 96 \]

### Step 3: Calculate the ratio \( p:q \)

Now that we have \( p = 120 \) and \( q = 96 \), we can find the ratio:
\[ p:q = 120:96 \]
To simplify this ratio, we divide both sides by 24:
\[ p:q = \frac{120}{24} : \frac{96}{24} = 5:4 \]

### Final Answer
The ratio \( p:q \) is \( 5:4 \).

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