Mock AIME 3 Pre 2005 Problems/Problem 2

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 ${8 + 7 \choose 7} = {15 \choose 7} = 6435 \equiv \boxed{435} \pmod{1000}$ .

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

Page

Toolbox

Search

Mock AIME 3 Pre 2005 Problems/Problem 2

Problem

Solution

See Also