5-digit number are formed using 2,3,5,7,9 with out repeating the digits. If p is the number of such numbers that exceeds 20000 and q be the number of those that lie between 30000 and 90000, then p:q is

Options

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

Answer

To solve the problem, we need to calculate the values of \( p \) and \( q \) as defined in the question.

### Step 1: Calculate \( p \)

**Explanation**: We need to find the total number of 5-digit numbers that can be formed using the digits 2, 3, 5, 7, and 9, which exceed 20,000. Since all the digits are greater than 2, any arrangement of these digits will exceed 20,000.

**Calculation**:
- The total number of 5-digit numbers that can be formed using the digits 2, 3, 5, 7, and 9 without repetition is given by \( 5! \) (5 factorial).

\[ p = 5! = 120 \]

### Step 2: Calculate \( q \)

**Explanation**: We need to find the total number of 5-digit numbers that lie between 30,000 and 90,000. This means the first digit can be either 3, 5, or 7 (since starting with 9 would exceed 90,000).

**Calculation**:
- If the first digit is 3, we can use the remaining digits (2, 5, 7, 9) to fill the next 4 positions. The number of arrangements is \( 4! \).
- If the first digit is 5, we can again use the remaining digits (2, 3, 7, 9) to fill the next 4 positions. The number of arrangements is \( 4! \).
- If the first digit is 7, we can use the remaining digits (2, 3, 5, 9) to fill the next 4 positions. The number of arrangements is \( 4! \).

Thus, the total number of valid arrangements for \( q \) is:

\[ q = 4! + 4! + 4! = 3 \times 4! = 3 \times 24 = 72 \]

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

**Calculation**:
Now, we can find the ratio \( p:q \):

\[ \frac{p}{q} = \frac{120}{72} = \frac{5}{3} \]

### Final Answer:
The ratio \( p:q \) is \( \frac{5}{3} \).