# How many entries...

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Each $\bullet$ in the mileage table below represents an entry indicating the distance between a pair of the five cities.If the table were extended to represent the distances between all pairs of 30 cities and each distance were to be represented by only one entry, how many entries would the table then have?

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A. 60
B. 435
C. 450
D. 465
E. 900
asked 4 years ago (2,970 points)

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ANS: B 435 => (30!/(28!*2!))
answered 4 years ago (2,070 points)
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In a table with 30 cities, there are 30(30)=900 boxes for entries.
However, since a city does not need to have any entry for a distance to and from itself, 30 entries are not needed on the diagonal through the table.
Thus, the necessary number of entries is reduced to 900-30=870 entries.
Then, it is given that each pair of cities only needs one table entry, not two as the table allows ;
therefore, the table only needs to have $\frac {870}{2}=435$ entries.
answered 4 years ago by (9,820 points)

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