# 1993 AJHSME Problems/Problem 22

## Problem

Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits? $\text{(A)}\ 22 \qquad \text{(B)}\ 99 \qquad \text{(C)}\ 112 \qquad \text{(D)}\ 119 \qquad \text{(E)}\ 199$

## Solution

There is $1$ two in the one-digit numbers.

The number of two-digit numbers with a two in the tens place is $10$ and the number with a two in the ones place is $9$. Thus the digit two is used $10+9=19$ times for the two digit numbers.

Now, Pat Peano only has $22-1-19=2$ remaining twos. You must subtract 1 because 22 is counted twice. The last numbers with a two that he can write are $102$ and $112$. He can continue numbering the last couple pages without a two until $120$, with the last number he writes being $\boxed{\text{(D)}\ 119}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 