$\qquad$ numbers greater than a million can be formed with the digits $2,3,0,3,4,2,3$.
$\qquad$ numbers greater than a million can be formed with the digits $2,3,0,3,4,2,3$.
60
360
420
120
Solution
All seven-digit numbers formed with the given digits are greater than a million.
Note that the digit at millions place cannot be 0 .
$\therefore \quad$ It can be any one of the digits $2,3,4$.
Case I :
Digit at millions place is ' 2 '.
Remaining 6 digits can be arranged in $\frac{6!}{3!}$ $=120$ ways.
Case II :
Digit at millions place is ' 3 '.
Remaining 6 digits can be arranged in $\frac{6!}{2!\times 2!}$ $=180$ ways.
Case III :
Digit at millions place is ' 4 '.
Remaining 6 digits can be arraigned in $\frac{6!}{3!\times 2!}$ $=60$ ways
$\begin{aligned} \therefore \quad \text { Required number of numbers } & =120+180+60 \\ & =360\end{aligned}$