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

Given the number to be formed by using the digits $0,2,3,4,6,7$ 

Now assuming counting integers starting from $2$

Case-I: if zero on $2^{\mathrm{nd}}$ place

(i) $2 0 2 \_$, now here at unit place $5$ number can take place except zero {as $2020<2022$}, so $5$ possible way.

(ii) $2 0 \_ \_$, now here at ten's place $4$ number $\left\{3,4,6,7\right\}$ can take place  and at unit place all $6$ number can take place so total $4\times 6=24$ possible way

Case-II: If non-zero number on $2^{\mathrm{nd}}$ place

i.e $2\underset{5}{ \underbrace{ \_}} \underset{6}{\underbrace{\_}}\underset{6}{ \underbrace{ \_}}$, here at hundredth place $5$ possibility will be there and at unit and ten's place $6 \& 6$ possibility will be there, so in total $5\times 6\times 6=180$ possible cases,

Case -III when counting integers starting from $3$

$\begin{array}{cccc}3& \underset{6}{\underbrace{\_}}& \underset{6}{\underbrace{\_}}& \underset{6}{\underbrace{\_}}\end{array}$ here total $6\times 6\times 6=216$ possible cases

Case -IV when counting integers starting from $4$

(i) If $0,2$ or $3$ on $2^{\mathrm{nd}}$ place

i.e., $\begin{array}{cccc}4& \underset{3}{\underbrace{\_}}& \underset{6}{\underbrace{\_}}& \underset{6}{\underbrace{\_}}\end{array}$ here total $3\times 6\times 6=108$ possible cases

(ii) If $4$ on $2^{\mathrm{nd}}$ place

i.e., $\begin{array}{cccc}3& 4& \underset{6}{\underbrace{\_}}& \underset{6}{\underbrace{\_}}\end{array}$ here total $6\times 6=36$ possible cases

So, total possible cases will be $5+24+180+216+108+36=569$ numbers