Difference between revisions of "2016 AMC 8 Problems/Problem 17"
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| + | == Problem == | ||
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An ATM password at Fred's Bank is composed of four digits from <math>0</math> to <math>9</math>, with repeated digits allowable. If no password may begin with the sequence <math>9,1,1,</math> then how many passwords are possible? | An ATM password at Fred's Bank is composed of four digits from <math>0</math> to <math>9</math>, with repeated digits allowable. If no password may begin with the sequence <math>9,1,1,</math> then how many passwords are possible? | ||
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<math>\textbf{(A)}\mbox{ }30\qquad\textbf{(B)}\mbox{ }7290\qquad\textbf{(C)}\mbox{ }9000\qquad\textbf{(D)}\mbox{ }9990\qquad\textbf{(E)}\mbox{ }9999</math> | <math>\textbf{(A)}\mbox{ }30\qquad\textbf{(B)}\mbox{ }7290\qquad\textbf{(C)}\mbox{ }9000\qquad\textbf{(D)}\mbox{ }9990\qquad\textbf{(E)}\mbox{ }9999</math> | ||
| − | ==Solution== | + | |
| − | For the first three digits, there are <math>10^3-1=999</math> combinations since <math>911</math> is not allowed. For the final digit, any of the <math>10</math> numbers are allowed. <math>999 \cdot 10 = 9990 \rightarrow \boxed{D}</math> | + | ==Solutions== |
| + | ===Solution 1=== | ||
| + | For the first three digits, there are <math>10^3-1=999</math> combinations since <math>911</math> is not allowed. For the final digit, any of the <math>10</math> numbers are allowed. <math>999 \cdot 10 = 9990 \rightarrow \boxed{\textbf{(D)}\ 9990}</math>. | ||
| + | |||
| + | ~CHECKMATE2021 | ||
| + | |||
| + | ===Solution 2=== | ||
| + | Counting the prohibited cases, we find that there are 10 of them. This is because, when we start with 9,1, and 1, we can have any of the 10 digits for the last digit. So, our answer is <math>10^4-10=\boxed{\textbf{(D)}\ 9990}.</math> | ||
| + | |||
| + | ~CHECKMATE2021 | ||
| + | |||
| + | ==Video Solution 1 (HOW TO THINK CREATIVELY!!!)== | ||
| + | https://youtu.be/bP2698xypP0 | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/slvWHYXz-20 | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==See Also== | ||
{{AMC8 box|year=2016|num-b=16|num-a=18}} | {{AMC8 box|year=2016|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 20:44, 21 January 2024
Contents
Problem
An ATM password at Fred's Bank is composed of four digits from 
to 
, with repeated digits allowable. If no password may begin with the sequence 
then how many passwords are possible?
Solutions
Solution 1
For the first three digits, there are 
combinations since 
is not allowed. For the final digit, any of the 
numbers are allowed. 
.
~CHECKMATE2021
Solution 2
Counting the prohibited cases, we find that there are 10 of them. This is because, when we start with 9,1, and 1, we can have any of the 10 digits for the last digit. So, our answer is 
~CHECKMATE2021
Video Solution 1 (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
| 2016 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
