# 2007 iTest Problems/Problem 33

## Problem

How many $\textit{odd}$ four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?

## Solution

Use casework to count the total possibilities. Note that order of picking numbers does not matter because there is only one way to arrange a set of digits to decreasing order.

• If the number ends in $1$, then there are $8$ remaining digits to choose from. Picking $3$ results in $\binom{8}{3} = 56$ possibilities.
• If the number ends in $3$, then there are $6$ remaining digits to choose from. Picking $3$ results in $\binom{6}{3} = 20$ possibilities.
• If the number ends in $5$, then there are $4$ remaining digits to choose from. Picking $3$ results in $\binom{4}{3} = 4$ possibilities.
• If the number ends in $7$ or $9$, then there are less than $3$ remaining digits to choose from, so there are no possibilities in that case.

Adding up the cases results in a total of $56+20+4 = \boxed{80}$ possibilities.