2007 iTest Problems/Problem 33

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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.

See Also

2007 iTest (Problems)
Preceded by:
Problem 32
Followed by:
Problem 34
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