1998 AHSME Problem 24

Probability Level 2

Call a 7-digit telephone number $d_1d_2d_3-d_4d_5d_6d_7$ memorable if the prefix sequence $d_1d_2d_3$ is exactly the same as either of the sequences $d_4d_5d_6$ or $d_5d_6d_7$ (possibly both). Assuming that each $d_i$ can be any of the ten decimal digits $0,1,2, \ldots 9$ , the number of difference memorable telephone numbers is

20000 19910 20100 19990

1 solution

Kevin Bourrillion
Apr 25, 2014

Each of the thousand possible three-digit prefixes has ten four-digit extensions that use it as the first three digits, and another ten that use it as the last three digits. 1000 x 20 = 20000. The only numbers this overcounts are those made up entirely of a single digit, like 333-3333, of which there are 10. 20000 - 10 = 19990.

1998 AHSME Problem 24

1 solution

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