A child has 3 distinct marbles and 4 distinct pockets. In how many ways can the marbles be distributed among the pockets (pockets may be empty)?

Introduction / Context:This is a classic “balls into boxes” scenario with distinct balls (marbles) and distinct boxes (pockets). Each marble independently chooses a pocket; pockets may be empty.

Given Data / Assumptions:

• Marbles are distinct (e.g., different colours).
• Pockets are distinct (different locations).
• Multiple marbles can go into the same pocket; empty pockets are allowed.

Concept / Approach:For each of the 3 marbles, choose one of 4 pockets. By the multiplication principle, total outcomes = 4 * 4 * 4 = 4^3.

Step-by-Step Solution:Choices per marble = 4.Marbles act independently: total = 4^3 = 64.

Verification / Alternative check:Equivalent to functions from a 3-element marble set to a 4-element pocket set; the number of functions is 4^3.

Why Other Options Are Wrong:12 and 60 correspond to permutations/arrangements; 256 is 4^4 (as if 4 marbles).

Common Pitfalls:Misinterpreting marbles as identical (which would be a stars-and-bars count) or forbidding empty pockets.

Final Answer:64