THE 68th ROMANIAN MATHEMATICAL OLYMPIAD

Since $84:(2\cdot 1\cdot 7)=6$, we distinguish the following cases:

(1) The digits are: $2,0,1,7,0,6$. If the first digit is $1$, the other $5$ digits can be arranged in $(5\cdot 4\cdot 3\cdot 2\cdot 1):2=60$ ways. Similar, if the first digit is $2$, $6$ or $7$. Thus we find $240$ numbers in this case.

(2) The digits are: $2,0,1,7,1,6$, If the first digit is $2$, the other $5$ digits can be arranged in $60$. Similar, if the first digit is $6$ or $7$. If the first digit is $1$, we obtain $5\cdot 4\cdot 3\cdot 2\cdot 1=120$ numbers. So there are $300$ numbers in this case.

(3) The digits are: $2,0,1,7,2,3$. If the first digit is $1$, $3$ or $7$, we obtain $3\cdot 60=180$ numbers, and if the first digit is $2$, we obtain $120$ numbers. Hence there are $300$ numbers in this case.

The number of numbers satisfying the conditions is thus $240+300+300=840$.