To determine the number of unique passwords the machine can generate under the given constraints, let's go step-by-step:

Step 1: List the available characters

1. Alphabets: Since only vowels are allowed, the available letters are $A, E, I, O, U$, giving us 5 possible choices.
2. Numbers: The single-digit natural numbers range from 1 to 9, so there are 9 possible choices.

Step 2: Determine the password structure

• The password is 5 characters long.
• It must contain exactly 2 alphabets and 3 numbers.
• Additional constraints:
  • The last 2 characters must contain at least one number.
  • The first 2 characters must contain at least one alphabet.
  • No characters can repeat.

Step 3: Calculate possibilities for the alphabets

• Choose 2 vowels out of 5. The number of ways to select 2 vowels is:

  $\binom{5}{2} = 10$

• Once chosen, the 2 vowels can be arranged in any order within the password. There are 2! ways to arrange the selected vowels:

  $2! = 2$

Step 4: Calculate possibilities for the numbers

• Choose 3 numbers out of 9. The number of ways to select 3 numbers is:

  $\binom{9}{3} = 84$

• Once chosen, the 3 numbers can be arranged in any order within the password. There are 3! ways to arrange the selected numbers:

  $3! = 6$

Step 5: Arranging alphabets and numbers in the password

Now, we need to arrange the 2 chosen alphabets and 3 chosen numbers to form a 5-character password with the constraints:

1. The first 2 positions must contain at least one alphabet.
2. The last 2 positions must contain at least one number.

Case Analysis

Let's analyze possible distributions that satisfy the conditions:

1. Case 1: (Alphabet, Alphabet, Number, Number, Number)
   • Choose the positions for the 2 alphabets out of 5: $\binom{5}{2} = 10$ ways.

Total arrangements:

• Choosing alphabets: $\binom{5}{2} \cdot 2! = 10 \cdot 2 = 20$ ways.
• Choosing numbers: $\binom{9}{3} \cdot 3! = 84 \cdot 6 = 504$ ways.

Thus, total unique passwords for this case:

$20 \cdot 504 = 10,080$

Step 6: Conclusion

The total number of unique passwords that can be generated by the machine is 10,080.

Do you want details or have any questions? Here are five related questions for further exploration:

1. What if consonants were also allowed, how would the calculation change?
2. If the password had to be exactly 6 characters instead, what would be the total combinations?
3. What would happen if repetition of characters were allowed?
4. How does the constraint on the position of alphabets and numbers impact the calculation?
5. If passwords were case-sensitive, how many combinations would that add?

Tip: Always identify constraints first, as they can significantly simplify complex problems by reducing the number of valid configurations.