To solve this, we calculate the total number of possible passwords and subtract cases that don’t satisfy the given conditions. Here’s the breakdown:

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Total possible passwords

The password has 8 characters, each of which can be one of 36 possible choices (26 letters + 10 digits).
$36^8 = 2,821,109,907,456$

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(a) Passwords with at least two letters

1. Total passwords: $36^8$

2. Passwords with fewer than 2 letters:

   • Case 1: No letters (all digits): $10^8$
   • Case 2: Exactly 1 letter:
     • Choose 1 position for the letter: $\binom{8}{1}$
     • Choose the letter: $26$
     • Fill remaining 7 positions with digits: $10^7$
       $\binom{8}{1} \cdot 26 \cdot 10^7 = 8 \cdot 26 \cdot 10^7 = 2,080,000,000$
   • Total passwords with fewer than 2 letters: $10^8 + 2,080,000,000 = 3,080,000,000$

3. Passwords with at least two letters:
   $36^8 - (10^8 + 2,080,000,000) = 2,818,029,907,456$

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(b) Passwords with at least two numbers

1. Total passwords: $36^8$

2. Passwords with fewer than 2 numbers:

   • Case 1: No numbers (all letters): $26^8$
   • Case 2: Exactly 1 number:
     • Choose 1 position for the number: $\binom{8}{1}$
     • Choose the number: $10$
     • Fill remaining 7 positions with letters: $26^7$
       $\binom{8}{1} \cdot 10 \cdot 26^7 = 8 \cdot 10 \cdot 26^7 = 873,218,688$
   • Total passwords with fewer than 2 numbers: $26^8 + 873,218,688 = 217,678,233,088 + 873,218,688 = 218,551,451,776$

3. Passwords with at least two numbers:
   $36^8 - (26^8 + 873,218,688) = 2,821,109,907,456 - 218,551,451,776 = 2,602,558,455,680$

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(c) Passwords with at least two letters and two numbers

1. Total passwords: $36^8$

2. Passwords not meeting the requirement:

   • Case 1: Fewer than 2 letters (calculated in part (a)): $10^8 + 2,080,000,000$
   • Case 2: Fewer than 2 numbers (calculated in part (b)): $26^8 + 873,218,688$
   • Double-counted case (fewer than 2 letters AND fewer than 2 numbers):
     • At most 1 letter and 1 number:
       • Choose 1 position for the letter: $\binom{8}{1} \cdot 26$
       • Choose 1 position for the number: $\binom{7}{1} \cdot 10$
       • Fill the remaining 6 positions with letters OR digits: $2^6 = 64$
         $\binom{8}{1} \cdot 26 \cdot \binom{7}{1} \cdot 10 \cdot 64 = 8 \cdot 26 \cdot 7 \cdot 10 \cdot 64 = 929,280$ $10^8 + 2,080,000,000 + 26^8 + 873,218,688 - 929,280$

3. Final calculation:

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