Difference between revisions of "2007 iTest Problems/Problem 33"
(Created page with "== Problem == 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 ==") |
Rockmanex3 (talk | contribs) (Solution to Problem 33) |
||
Line 3: | Line 3: | ||
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 03:29, 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.
Adding up the cases results in a total of possibilities.
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 32 |
Followed by: Problem 34 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4 |