# Difference between revisions of "2006 AMC 10A Problems/Problem 21"

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== Problem == | == Problem == | ||

− | How many four-digit positive | + | How many four-[[digit]] [[positive integer]]s have at least one digit that is a 2 or a 3? |

<math>\mathrm{(A) \ } 2439\qquad\mathrm{(B) \ } 4096\qquad\mathrm{(C) \ } 4903\qquad\mathrm{(D) \ } 4904\qquad\mathrm{(E) \ } 5416\qquad</math> | <math>\mathrm{(A) \ } 2439\qquad\mathrm{(B) \ } 4096\qquad\mathrm{(C) \ } 4903\qquad\mathrm{(D) \ } 4904\qquad\mathrm{(E) \ } 5416\qquad</math> | ||

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== Solution == | == Solution == | ||

+ | Since we are asked for the number of positive 4-digit [[integer]]s with at least 2 or 3 in it, we can find this by finding the total number of 4-digit integers and subtracting off those which do not have any 2s or 3s as digits. | ||

− | + | The total number of 4-digit integers is <math>9 \cdot 10 \cdot 10 \cdot 10 = 9000</math>, since we have 10 choices for each digit except the first (which can't be 0). | |

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− | |||

− | + | Similarly, the total number of 4-digit integers without any 2 or 3 is <math>7 \cdot 8 \cdot 8 \cdot 8 = 3584</math>. | |

− | Therefore, the total number of positive 4-digit integers that have at least one 2 or 3 in | + | Therefore, the total number of positive 4-digit integers that have at least one 2 or 3 in their [[decimal representation]] is <math>9000-3584=5416 \Longrightarrow \mathrm{(E)} </math> |

== See Also == | == See Also == |

## Revision as of 14:58, 16 October 2006

## Problem

How many four-digit positive integers have at least one digit that is a 2 or a 3?

## Solution

Since we are asked for the number of positive 4-digit integers with at least 2 or 3 in it, we can find this by finding the total number of 4-digit integers and subtracting off those which do not have any 2s or 3s as digits.

The total number of 4-digit integers is , since we have 10 choices for each digit except the first (which can't be 0).

Similarly, the total number of 4-digit integers without any 2 or 3 is .

Therefore, the total number of positive 4-digit integers that have at least one 2 or 3 in their decimal representation is