# FMM19

### From charlesreid1

# Friday Morning Math Problem

## Radical Sums

Call a seven-digit phone number D1 D2 D3 - D4 D5 D6 D7 **memorable** if the prefix sequence D1 D2 D3 is exactly the same as either of the sequences D4 D5 D6 or D5 D6 D7 (possibly both). Assuming that each Di can be any of the ten decimal digits 0, 1, 2, ..., 9, find the number of different memorable telephone numbers.

Solution |
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There are 10,000 ways to write the last four digits D4 D5 D6 D7. Among those are 10,000 - 10 = 9,990 for which not all the digits are the same. For each of these, there are exactly two ways to adjoin the three digits D1 D2 D3 to obtain a memorable number. There are ten memorable numbers for which the last four digits are the same, for a total of
Alternative solution: Let A denote the set of telephone numbers for which D1 D2 D3 is the same as D4 D5 D6. Let B denote the set of telephone numbers for which D1 D2 D3 coincides with D5 D6 D7. A telephone number belongs to if and only if D1 = D4 = D5 = D2 = D6 = D3 = D7, so . Thus, by the Inclusion Exclusion Principle, |

## Flags

Friday Morning Math ProblemsSpider Socks and Shoes FMM1 Sums of Powers of 2 FMM2 Fifty Coins FMM3 The Zeta Monogram FMM4 The Cthulhus Monogram FMM4B Multiplication Logic FMM5 The Termite and the Cube FMM6 Sharing Dump Trucks FMM7 The Flippant Juror FMM8 Bus Routes FMM9 A Robust Bus System FMM9B Square-Free Sequence FMM10 Inferring Rule from Sequence FMM11 Checkerboard Color Schemes FMM12 One-Handed Chords FMM13 First Ace FMM14 Which Color Cab FMM15 Petersburg Paradox Revisited FMM16 A Binomial Challenge FMM17 A Radical Sum FMM18 Memorable Phone Numbers FMM19 Arrange in Order FMM20 A Pair of Dice Games: FMM21
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