   SEARCH HOME Math Central Quandaries & 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.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.