The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ___________.

<li><p><strong>Total ways without any restrictions :</strong> <br/><br/>There are $3^{20}$ ways to distribute the oranges to the 3 children.</p> </li> <li><p><strong>Number of ways one child receives no orange :</strong> <br/><br/>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&#39;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) $</p> </li> <li><p><strong>Number of ways two children receive no orange :</strong> <br/><br/>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 $</p> </li> </ol> <br/>Number of ways <br/><br/>$=$ Total $-($ One child receive no orange + two child receive no orange) <br/><br/>$$ \begin{aligned} & =3^{20}-\left({ }^3 C_1\left(2^{20}-2\right)+{ }^3 C_2 1^{20}\right) \\\\ & =3483638676 \end{aligned} $$