How many of the natural numbers can be formed from the digits 2, 3, 5, 7, 9 ? How many of them are odd?

Answer

To solve the problem of how many natural numbers can be formed from the digits 2, 3, 5, 7, and 9, we will consider two cases: when digits can repeat and when digits cannot repeat.

### Case 1: Digits Cannot Repeat

1. **1-Digit Numbers**:
- The available digits are 2, 3, 5, 7, and 9.
- Total options = 5 (2, 3, 5, 7, 9).

2. **2-Digit Numbers**:
- For the first digit, we have 5 options (2, 3, 5, 7, 9).
- For the second digit, since digits cannot repeat, we have 4 options left.
- Total = 5 × 4 = 20.

3. **3-Digit Numbers**:
- For the first digit, we have 5 options.
- For the second digit, we have 4 options.
- For the third digit, we have 3 options left.
- Total = 5 × 4 × 3 = 60.

4. **4-Digit Numbers**:
- For the first digit, we have 5 options.
- For the second digit, we have 4 options.
- For the third digit, we have 3 options.
- For the fourth digit, we have 2 options left.
- Total = 5 × 4 × 3 × 2 = 120.

5. **5-Digit Numbers**:
- For the first digit, we have 5 options.
- For the second digit, we have 4 options.
- For the third digit, we have 3 options.
- For the fourth digit, we have 2 options.
- For the fifth digit, we have 1 option left.
- Total = 5 × 4 × 3 × 2 × 1 = 120.

6. **Total Natural Numbers (Digits Cannot Repeat)**:
- Total = 5 (1-digit) + 20 (2-digit) + 60 (3-digit) + 120 (4-digit) + 120 (5-digit)
- Total = 5 + 20 + 60 + 120 + 120 = 325.

### Case 2: Digits Can Repeat

1. **1-Digit Numbers**:
- The available digits are still 2, 3, 5, 7, and 9.
- Total options = 5.

2. **2-Digit Numbers**:
- For the first digit, we have 5 options.
- For the second digit, since digits can repeat, we again have 5 options.
- Total = 5 × 5 = 25.

3. **3-Digit Numbers**:
- For the first digit, we have 5 options.
- For the second digit, we have 5 options.
- For the third digit, we have 5 options.
- Total = 5 × 5 × 5 = 125.

4. **4-Digit Numbers**:
- For each digit, we have 5 options.
- Total = 5 × 5 × 5 × 5 = 625.

5. **5-Digit Numbers**:
- For each digit, we have 5 options.
- Total = 5 × 5 × 5 × 5 × 5 = 3125.

6. **Total Natural Numbers (Digits Can Repeat)**:
- Total = 5 (1-digit) + 25 (2-digit) + 125 (3-digit) + 625 (4-digit) + 3125 (5-digit)
- Total = 5 + 25 + 125 + 625 + 3125 = 3905.

### Summary of Results
- Total natural numbers formed when digits cannot repeat: **325**.
- Total natural numbers formed when digits can repeat: **3905**.