<s>[INST] Number of ways to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball 
How many ways are there to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball?

My approach is $ \binom{5}{3} 3! $ + $ \binom{2}{2} \binom{3}{2}2!$ which is equal to $66$?
 [/INST] $$\sum_{r_1+r_2+r_3=5,r_i\geq1}\binom{5}{r_1}\binom{5-r_1}{r_2}\binom{5-r_1-r_2}{r_3}=$$
$$=\sum_{r_1+r_2+r_3=5,r_i\geq1}\frac{5!}{r_1!r_2!r_3!}=\frac{5!}{1!1!3!}+\frac{5!}{1!3!1!}+\frac{5!}{3!1!1!}+\frac{5!}{1!2!2!}+\frac{5!}{2!1!2!}+\frac{5!}{2!2!1!}=$$
$$=3(20+30)=150$$
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