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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. How many telephone numbers have all six digits distinct?
Solution
If first two digits is 41, then the remaining 4 digits can be arranged in 8P4 ways
= `(8!)/((8 - 4)!)`
= `(8!)/(4!)`
= `(8 xx 7 xx 6 xx 5 xx 4!)/(4!)`
= 1680
Similarly, first two digits can be 42 or 46 or 62 or 64.
∴ Total number of telephone numbers have all digits distinct = 5 × 1680 = 8400
Hence, the required telephone numbers = 8400
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