IIT-JEE 2012 · Paper 1, Mathematics question 41

Onto functions from 5 distinct balls to 3 persons, by inclusion-exclusion:
$$3^5-\binom312^5+\binom321^5=243-96+3=150.$$

Check by splits.

• Sizes 3, 1, 1: choose the trio ($\binom53=10$), its owner (3), and give the two single balls to the other two persons (2): 60.
• Sizes 2, 2, 1: choose the single ball (5), pair up the other four (3 ways), and assign the three groups to persons ($3!=6$): 90.

$60+90=150$.

(B)