1998 AHSME Problems/Problem 24

Problem

Call a $7$ -digit telephone number $d_1d_2d_3-d_4d_5d_6d_7$ memorable if the prefix sequence $d_1d_2d_3$ is exactly the same as either of the sequences $d_4d_5d_6$ or $d_5d_6d_7$ (possibly both). Assuming that each $d_i$ can be any of the ten decimal digits $0,1,2, \ldots, 9$ , the number of different memorable telephone numbers is

Let $A$ represent the set of telephone numbers with $\overline{d_1d_2d_3} = \overline{d_4d_5d_6}$ (of which there are $1000$ possibilities for $\overline{d_1d_2d_3}$ and $10$ for $d_7$ ), and $B$ those such that $\overline{d_1d_2d_3} = \overline{d_5d_6d_7}$ . Then $A \cap B$ (the telephone numbers that belong to both $A$ and $B$ ) is the set of telephone numbers such that $d_1 = d_2 = d_3 = d_4 = d_5 = d_6 = d_7$ , of which there are $10$ possibilities. By the Principle of Inclusion-Exclusion,