The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3

Question

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $0,2,3,4,6,7$ is

Professor PC Thomas Classes · Answer

(569) : $\underline{a} \quad \underline{b} \quad \underline{c} \quad \underline{d}$ Case-I : $a=2, b=0, c=2, d$ can take values 2, 3, 4, 6, 7\ $ightarrow$ so 5 numbers\ $a=2, b=0, c$ can take values $3,4,6,7, d$ has 6 choices\ $ightarrow$ so 24 numbers\ Case-II : $a=2, b$ can take values $2,3,4,6,7$, $c$ and $d$ have 6 choices each $ightarrow$ so $5 \cdot 6 \cdot 6=180$ numbers\ Case-III : $a=3, b, c, d$ each has 6 choices $ightarrow$ so $1 \cdot 6^{3}=216$ numbers\ Case-IV : $a=4, b=0,2,3,4, c$ and $d$ have 6 choices each\ $ightarrow$ so $4 \cdot 6^{2}=144$ numbers\ The total number of 4 digit integers\ $=5+24+180+216+144=569$\