Difference between revisions of "2005 AMC 10B Problems/Problem 18"
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All of David's telephone numbers have the form <math>555-abc-defg</math>, where <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, <math>e</math>, <math>f</math>, and <math>g</math> are distinct digits and in increasing order, and none is either <math>0</math> or <math>1</math>. How many different telephone numbers can David have? | All of David's telephone numbers have the form <math>555-abc-defg</math>, where <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, <math>e</math>, <math>f</math>, and <math>g</math> are distinct digits and in increasing order, and none is either <math>0</math> or <math>1</math>. How many different telephone numbers can David have? | ||
| − | <math>\ | + | <math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9 </math> |
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== Solution == | == Solution == | ||
| − | The only digits available to use in the phone number are <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, and <math>9</math>. There are only <math>7</math> spots left among the <math>8</math> numbers, so we need to find the number of ways to choose <math>7</math> numbers from <math>8</math>. The answer is | + | The only digits available to use in the phone number are <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, and <math>9</math>. There are only <math>7</math> spots left among the <math>8</math> numbers, so we need to find the number of ways to choose <math>7</math> numbers from <math>8</math>. The answer is then <math>\dbinom{8}{7}=\dfrac{8!}{7!\,(8-7)!}=\boxed{\textbf{(D) } 8}</math> |
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| + | Alternatively, we could just choose <math>1</math> out of the <math>8</math> numbers not to be used. There are obviously <math>\boxed{8}</math> ways to do so. | ||
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| + | Note: for each combination of 7 numbers, exactly 1 is in increasing order -Williamgolly | ||
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== See Also == | == See Also == | ||
{{AMC10 box|year=2005|ab=B|num-b=17|num-a=19}} | {{AMC10 box|year=2005|ab=B|num-b=17|num-a=19}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 14:33, 16 December 2021
Problem
All of David's telephone numbers have the form 
, where 
, 
, 
, 
, 
, 
, and 
are distinct digits and in increasing order, and none is either 
or 
. How many different telephone numbers can David have?
Solution
The only digits available to use in the phone number are 
, 
, 
, 
, 
, 
, 
, and 
. There are only spots left among the numbers, so we need to find the number of ways to choose numbers from . The answer is then 
Alternatively, we could just choose out of the numbers not to be used. There are obviously 
ways to do so.
Note: for each combination of 7 numbers, exactly 1 is in increasing order -Williamgolly
See Also
| 2005 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 AMC 10 Problems and Solutions | ||
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