Total ways without any restrictions :

There are $3^{20}$ ways to distribute the oranges to the 3 children.

Number of ways one child receives no orange :

Choose 1 child out of the 3 to not receive any orange in ${ }^3 C_1 = 3$ ways. Distribute 20 oranges to the remaining 2 children in $2^{20}$ ways. However, we've included the scenarios where the 2 children each get all the oranges. So, we subtract the 2 ways where one of the two remaining children gets all the oranges. $ { }^3 C_1(2^{20} - 2) $

Number of ways two children receive no orange :

Choose 2 children out of the 3 to not receive any oranges in ${ }^3 C_2 = 3$ ways. The third child will receive all 20 oranges in $1^{20} = 1$ way. $ { }^3 C_2 \times 1^{20} = 3 $