Difference between revisions of "2007 iTest Problems/Problem 33"
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Rockmanex3 (talk | contribs) (Solution to Problem 33) |
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How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? | How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? | ||
| − | == Solution == | + | ==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 <math>1</math>, then there are <math>8</math> remaining digits to choose from. Picking <math>3</math> results in <math>\binom{8}{3} = 56</math> possibilities. | ||
| + | * If the number ends in <math>3</math>, then there are <math>6</math> remaining digits to choose from. Picking <math>3</math> results in <math>\binom{6}{3} = 20</math> possibilities. | ||
| + | * If the number ends in <math>5</math>, then there are <math>4</math> remaining digits to choose from. Picking <math>3</math> results in <math>\binom{4}{3} = 4</math> possibilities. | ||
| + | * If the number ends in <math>7</math> or <math>9</math>, then there are less than <math>3</math> remaining digits to choose from, so there are no possibilities in that case. | ||
| + | |||
| + | Adding up the cases results in a total of <math>56+20+4 = \boxed{80}</math> possibilities. | ||
| + | |||
| + | ==See Also== | ||
| + | {{iTest box|year=2007|num-b=32|num-a=34}} | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
Latest revision as of 02:29, 14 June 2018
Problem
How many 
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
, then there are 
remaining digits to choose from. Picking 
results in 
possibilities. - If the number ends in , then there are
remaining digits to choose from. Picking results in 
possibilities. - If the number ends in
, then there are 
remaining digits to choose from. Picking results in 
possibilities. - If the number ends in
or 
, then there are less than remaining digits to choose from, so there are no possibilities in that case.
, then there are 
remaining digits to choose from. Picking 
results in 
possibilities.
remaining digits to choose from. Picking 
possibilities.
, then there are 
remaining digits to choose from. Picking 
possibilities.
or 
, then there are less than Adding up the cases results in a total of 
possibilities.
See Also
| 2007 iTest (Problems, Answer Key) | ||
| Preceded by: Problem 32 |
Followed by: Problem 34 | |
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