Question

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

• A. $8400$
• B. $7200$
• C. $9200$
• D. None of these

Answer 1

The correct option is A $8400$

Suppose that the first two digits are 41, remaining 4 are to be chosen from 0,2,3,5,6,7,8,9 so as to make all the digits distinct.

Thus remaining 4 can be chosen in ${}^8{P}_4$ ways. We have 8 letters and they are to be arranged in 8 different ways.

First place can be arranged in 8 ways, 2nd in 7 ways, 3rd in 6 and 4th in 5 different ways.

Thus number of such telephone numbers$=8\times 7\times 6\times 5=1680$

Now considering the all given first two digits (41 or 42 or 46 or 64) , total number of telephone numbers$=1680\times 5=8400$