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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?

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If first two digit is 41, the remaining 4 digits can be arranged in ^8P_4  ways= 8*7*6*5 =1680

In addition, if first two digits are 42, 46, 62 or 64 the remaining 4 can be arranged in ^8P_4  ways=8*7*6*5 =1680 ways.

So, total number of telephone numbers having all 6 digits different = 5*1680 =8400

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