A bank asked its customers to create a personal six-digit password, formed only by digits from 0 to 9, for access to their checking account via the internet. However, an expert in electronic security systems recommended to the bank's management to re-register its users, requesting, for each one of them, the creation of a new six-digit password, now allowing the use of the 26 letters of the alphabet, in addition to digits from 0 to 9. In this new system, each uppercase letter was considered distinct from its lowercase version. Furthermore, the use of other types of characters was prohibited. One way to evaluate a change in the password system is to verify the improvement coefficient, which is the ratio of the new number of password possibilities to the old one. The improvement coefficient of the recommended change is (A) $\frac{62^{6}}{10^{6}}$ (B) $\frac{62!}{10!}$ (C) $\frac{62! \cdot 4!}{10! \cdot 56!}$ (D) $62! - 10!$ (E) $62^{6} - 10^{6}$
A bank asked its customers to create a personal six-digit password, formed only by digits from 0 to 9, for access to their checking account via the internet.
However, an expert in electronic security systems recommended to the bank's management to re-register its users, requesting, for each one of them, the creation of a new six-digit password, now allowing the use of the 26 letters of the alphabet, in addition to digits from 0 to 9. In this new system, each uppercase letter was considered distinct from its lowercase version. Furthermore, the use of other types of characters was prohibited.
One way to evaluate a change in the password system is to verify the improvement coefficient, which is the ratio of the new number of password possibilities to the old one.
The improvement coefficient of the recommended change is
(A) $\frac{62^{6}}{10^{6}}$
(B) $\frac{62!}{10!}$
(C) $\frac{62! \cdot 4!}{10! \cdot 56!}$
(D) $62! - 10!$
(E) $62^{6} - 10^{6}$