In a certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64. The number of telephone numbers having all six digits distinct is

Suppose, the first two digits are 41, remaining four digits are to be chosen from 0, 2, 3, 5, 6, 7, 8, 9 so as to make all the digits distinct.

∴ Remaining four can be chosen in P (8, 4) ways.

We have 8 letters and they are to be arranged at 4 places.

First place can be arranged in 8 different ways.

Second place can be arranged in 7 different ways.

Third place can be arranged in 6 different ways.

Fourth place can be arranged in 5 different ways.

Number of such telephone numbers = 8 × 7 × 6 × 5 = 1680.

Now, considering the all given first two digits (41 or 42 or 46 or 64), total number of the telephone numbers

= 1680 × 5 = 8400.