Bunuel wrote:
A city's telephone numbers are made up of 6 digits. Knowing that the first digit can never be zero, if the telephone numbers change to 7 digits, how many more telephone numbers could be created?
(A) 81 x 10^3
(B) 90 x 10^3
(C) 81 x 10^4
(D) 81 x 10^5
(E) 90 x 10^5
The problem can be solved in two ways:
A. If only 6 digits, including a '0', are used
B. All digits are used.
B seems to be the case here, however.
A. 5 digits can take the 1st place and each of the rest 5 places can be taken in 6 ways.
Ways a telephone number can be formed = \(5*6^5\)
In case of a 7 digit number, total ways are \(5*6^6\)
Difference = \(5*6^6 - 5*6^5 = 5^2*6^5\)
B. 9 digits can take the 1st place and each of the rest 5 places can be taken in 10 ways.
Ways a telephone number can be formed = \(9*10^5\)
In case of a 7 digit number, total ways are \(9*10^6\)
Difference = \(9*10^6 - 9*10^5 = 9^2*10^5\)
Answer D.
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