Problem 9

Explanation

We need to form a code with the pattern Letter - Digit - Letter - Digit.

Let the slots for the code be:

Available sets:

• Vowels: $\{A, I, U, E, O\}$ ($5$ letters).
• Odd digits for $D_1$: $\{1, 3, 5, 7, 9\}$ ($5$ digits).
• Choice digits for $D_2$: $\{0, 1, 2, 4, 6\}$ ($5$ digits).

Rules and Constraints:

1. $L_1 \neq A$.
2. $L_2 = O$.
3. No repetition of letters or digits is allowed.

Since $D_1$ must be odd and $D_2$ is chosen from a specific set, there is an overlap at the digit $1$. We need to split the calculation into cases based on the value of $D_2$.

Case 1 If The Second Digit Is 1

If we choose the digit $1$ for position $D_2$:

1. $L_2$ (Second letter): Must be $O$ ($1$ way).
2. $D_2$ (Second digit): Must be $1$ ($1$ way).
3. $L_1$ (First letter): Vowel other than $A$ and other than $O$ (since $O$ is used in $L_2$).
   • Choices: $\{I, U, E\}$ ($3$ ways).
4. $D_1$ (First digit): Odd digit other than $1$ (since $1$ is used in $D_2$).
   • Choices: $\{3, 5, 7, 9\}$ ($4$ ways).

Total ways for Case $1$:

Case 2 If The Second Digit Is Not 1

If we choose a digit other than $1$ for position $D_2$ (which means it is even):

1. $L_2$ (Second letter): Must be $O$ ($1$ way).
2. $D_2$ (Second digit): Chosen from $\{0, 2, 4, 6\}$ ($4$ ways).
3. $L_1$ (First letter): Vowel other than $A$ and other than $O$.
   • Choices: $\{I, U, E\}$ ($3$ ways).
4. $D_1$ (First digit): Odd digit $\{1, 3, 5, 7, 9\}$. Since $D_2$ is even, there is no conflict. All odd digits can be chosen.
   • Choices: $5$ ways.

Total ways for Case $2$:

Total Combinations

Sum the results from both cases:

Therefore, the number of codes that can be formed is $72$.