Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 2"

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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>.  
 
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==
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==See Also==
 
{{Mock AIME box|year=Pre 2005|n=3|num-b=1|num-a=3}}
 
{{Mock AIME box|year=Pre 2005|n=3|num-b=1|num-a=3}}
  
 
[[Category:Intermediate Combinatorics Problems]]
 
[[Category:Intermediate Combinatorics Problems]]

Latest revision as of 10:16, 29 November 2019

Problem

Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.)

Solution

Note that a $7$ digit increasing integer is determined once we select a set of $7$ digits. To determine the number of sets of $7$ digits, consider $9$ urns labeled $1,2,\cdots,9$ (note that $0$ is not a permissible digit); then we wish to drop $7$ balls into these urns. Using the ball-and-urn argument, having $9$ urns is equivalent to $8$ dividers, and there are ${8 + 7 \choose 7} = {15 \choose 7} = 6435 \equiv \boxed{435} \pmod{1000}$.

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