Eight distinct toys are to be distributed among 5 distinct children. If each toy can be given to any child and children may receive any number of toys (including zero), in how many different ways can this distribution be done?

Introduction / Context:
This question is a distribution problem with distinct objects (toys) and distinct recipients (children). Each toy can go independently to any child, and there is no restriction on how many toys a child may get. It is a classic example of counting functions from one finite set to another.

Given Data / Assumptions:

• There are 8 distinct toys.
• There are 5 distinct children.
• Each toy must be given to exactly one child.
• A child can receive any number of toys, including zero.
• Different assignments of toys to children count as different distributions.

Concept / Approach:
Think of each toy as making an independent choice of which child it goes to. For each toy there are 5 possible children. The choices for different toys are independent, so the total number of distributions is the product of the choices at each step. That naturally leads to a power expression of the form 5^8.

Step-by-Step Solution:
Step 1: Consider the first toy. It can be given to any of the 5 children, so there are 5 choices.Step 2: Consider the second toy. Regardless of who received the first toy, this toy again has 5 choices of children.Step 3: The same logic applies to each of the 8 toys. Each toy has 5 possible recipients.Step 4: Because the assignment of each toy is independent of the others, the total number of distributions is 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5.Step 5: This product is 5^8.

Verification / Alternative check:
An alternative interpretation is to view each distribution as a function from the set of 8 toys to the set of 5 children, where each toy is mapped to exactly one child. The number of such functions from an 8 element set to a 5 element set is 5^8, which matches the counting argument above. You can also check with a small example, such as 2 toys and 3 children, where the correct answer is 3^2 = 9, and manually listing them will confirm the formula.

Why Other Options Are Wrong:

• 5P8 and 8P5: These are permutation counts and would apply if we were arranging items in order, not assigning each toy to a child.
• 8^5: This would count functions in the opposite direction, from 5 children to 8 toys, which is not the situation here.

Common Pitfalls:
One common error is to incorrectly use combinations or permutations instead of recognising this as a simple independent choice problem. Another is to believe that each child must receive at least one toy, which would add an extra constraint and change the counting method completely. Carefully reading that children may receive any number of toys, including zero, is crucial to using the 5^8 formula.

Final Answer:
The number of ways to distribute the 8 distinct toys among 5 children is 5^8.