Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 2"
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==Solution== | ==Solution== | ||
− | {{ | + | Note that a <math>7</math> digit increasing integer is determined once we select a set of <math>7</math> digits. To determine the number of sets of <math>7</math> digits, consider <math>9</math> urns labeled <math>1,2,\cdots,9</math> (note that <math>0</math> is not a permissible digit); then we wish to drop <math>7</math> balls into these urns. Using the ball-and-urn argument, having <math>9</math> urns is equivalent to <math>8</math> dividers, and there are <math>{8 + 7 \choose 7} = {15 \choose 7} = 6435 \equiv \boxed{435} \pmod{1000}</math>. |
==See also== | ==See also== |
Revision as of 20:41, 19 June 2008
Problem
Let denote the number of
digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when
is divided by
. (Repeated digits are allowed.)
Solution
Note that a digit increasing integer is determined once we select a set of
digits. To determine the number of sets of
digits, consider
urns labeled
(note that
is not a permissible digit); then we wish to drop
balls into these urns. Using the ball-and-urn argument, having
urns is equivalent to
dividers, and there are
.
See also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |