2004 AMC 10B Problems/Problem 2

Problem

How many two-digit positive integers have at least one $7$ as a digit?

$\mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 19 \qquad \mathrm{(D) \ } 20\qquad \mathrm{(E) \ } 30$

Solution

Ten numbers $(70,71,\dots,79)$ have $7$ as the tens digit. Nine numbers $(17,27,\dots,97)$ have it as the ones digit. Number $77$ is in both sets.

Thus the result is $10+9-1=18 \Rightarrow$ $\boxed{\mathrm{(B)}\ 18}$ .

Solution 2

We use complementary counting. The complement of having at least one $7$ as a digit is having no $7$ s as a digit.

Usually, we have $9$ digits to choose from for the first digit, and $10$ digits for the second. This gives a total of $90$ $2$ -digit numbers.

But since we cannot have a $7$ as a digit, we have $8$ first digits and $9$ second digits to choose from.

Thus there are $72$ $2$ -digit numbers without a $7$ as a digit.

$90$ (The total number of $2$ -digit numbers) - $72$ (The number of $2$ -digit numbers without a $7$ ) $= 18 \Rightarrow$ $\boxed{\mathrm{(B)}\18}$ (Error compiling LaTeX. Unknown error_msg).

2004 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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2004 AMC 10B Problems/Problem 2

Contents

Problem

Solution

Solution 2

See also