Math CentralQuandaries & Queries


Question from Vonda, a parent:

How many four digit numbers contain the digit pattern 85 at least once and only once?

Hi Vonda.

What's a "four digit number"? Is 0855 a four-digit number? If so, then you can answer the question this way:

There are two unknown digits. Each could be 0-9. That's ten possibilities. Since there are two of them, there are 102 possible values for xy. As well, the "85" could be in front of the xy, in between the x and the y or after the xy. That's three possible positions, so multiply by 3 and we have 3 x 102.

Finally, remove the "85" 8 5 and the 8 5 "85" that were in there (once due to adding after and once due to adding before: we don't want either of these two).

So the final answer is (3 x 102) - 2 = 298.

If you don't allow 0 for the leading digit (or 00 for the two leading digits, you will have to adjust the procedure for this.

Stephen La Rocque.

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