1. A telephone company decided to organize the procedure for issuing telephone numbers in a new microdistrict as follows: residents of apartments with even numbers will be given seven-digit numbers consisting only of even digits, and residents of apartments with odd numbers will be given seven-digit numbers consisting only of odd digits. How many total possibilities exist to form seven-digit numbers: a) only with even digits, not starting with 0, b) only with odd digits, c) which type of number will be more and by how many times?

(12 points)
Solution. a) Even digits are $0,2,4,6,8$, a seven-digit number cannot start with 0, so there can be 4 digits in the first place, and any of the 5 even digits can follow, the total number of combinations is $N_{\text {even }}=4 \cdot 5^{6}$, b) there are 5 odd digits: $1,3,5,7,9$ and any of them can stand in each of the 7 places, the total number of combinations is $N_{\text {odd }}=5^{7}$,

c) there are more odd numbers $\frac{N_{\text {odd }}}{N_{\text {even }}}=\frac{5^{7}}{4 \cdot 5^{6}}=\frac{5}{4}=1.25$.

Answer: a) 62500, b) 78125, c) there are 1.25 times more odd numbers.