To play the EuroMillions lottery, one selects five main numbers from $1$ to $50$ and two Lucky Stars numbered from $1$ to $11$.
I would like to know what the number of possibilities is given the following constraints:
- Constraint #1: one can not pick adjacent numbers (e.g., $1$ and $2$, or $7$ and $8$). This constraint applies both to main numbers and Lucky Star numbers.
- Constraint #2: one can pick at most two main numbers from each "decimal house" ($1$ to $10$, $11$ to $20$,...).
What is the number of possibilities of number sets knowing that one can only pick a number for each decimal house ?
If we have to choose one number from 1 to 10, one number from 11 to 20, one number from 21 to 30, one number from 31 to 40, one number from 41 to 50:
NUMBER OF POSSIBILITIES = 10 * 10 * 10 *10 *10 = 10^5