Four different prizes are to be given to 3 different students. A student may get any number of prizes, including none, and every prize must be given. In how many ways can the prizes be given?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 81
Each prize is given on its own, and it can go to any of the students; the choices for the prizes are multiplied.
Each of the 4 prizes can go to any of the 3 students: 3 choices for each prize
Ways \(=3\times3\times3\times3=3^4=81\)
Why the other options are incorrect
Option (1) 64 — Incorrect. This is \(4^3\), which gives each student a choice of prize; but a student may get several prizes or none, so the choice belongs to each prize.
Option (3) 24 — Incorrect. This is \(^4P_3\), the count if each student had to get exactly one prize and one prize stayed back.
Option (4) 12 — Incorrect. This is \(4\times3\), which is not a counting rule.