2002 AMC 12P Problems/Problem 7

The following problem is from both the 2002 AMC 12P #7 and 2002 AMC 10P #20, so both problems redirect to this page.

Problem

How many three-digit numbers have at least one $2$ and at least one $3$?

$\text{(A) }52 \qquad \text{(B) }54  \qquad \text{(C) }56 \qquad \text{(D) }58 \qquad \text{(E) }60$

Solution

We can do this problem with some simple case work.

Case 1: The hundreds place is not $2$ or $3.$ This means that the tens place and ones place must be $2$ and $3$ respectively or $3$ and $2$ respectively. This case covers $1, 4, 5, 6, 7, 8,$ and $9,$ so it gives us $2 \cdot 7 = 14$ cases.

Case 2: The hundreds place is $2.$ This means that $3$ must be in the tens place or ones place. Starting with cases in which the tens place is not $3$, we get $203, 213, 223, 243, 253, 263, 273, 283,$ and $293.$ With cases in which the tens place is $3$, we have $230-239$, or $10$ more cases. This gives us $9 + 10=19$ cases.

Case 3: The hundreds place is $3.$ This case is almost identical to the second case, just swap the $2$s with $3$s and $3$s with $2$s in the reasoning and its the same, giving us an additional $19$ cases.

Adding up all of these cases gives $14+19+19=52$ cases, or $\boxed{\textbf{(A) }52}.$

See also

2002 AMC 10P (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
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
2002 AMC 12P (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
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 12 Problems and Solutions

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