- A security code is made up of 4 numerical digits followed by 3 distinct uppercase letters.
Given that the digits must be from the set \(\{ 1,2,3,4,5 \}\) and the letters from the set \{A, B, C, D\}
- determine the total number of possible codes using this system.
To enable more codes to be generated, the system is adapted so that the 3 letters can appear anywhere in the code but no letter can be next to another letter.
- Determine the increase in the number of codes using this adapted system.
(ii) A combination lock code consists of four distinct digits that can be read as a positive integer, \(N = a b c d\), satisfying
- all the digits are odd
- \(\quad N\) is divisible by 9
- the digits appear in either ascending or descending order
- \(\quad N \equiv e ( \bmod a b )\) where \(a b\) is read as a two-digit number and \(e\) is the odd digit that is not used in the code
- Use the first two properties to determine the four digits used in the code.
- Hence determine the code on the lock.