Problem:
Five different games are to be distributed among four children randomly. The probability that each child gets at least one game is:
a) 1/4
b) 15/64
c) 21/64
d) none of these

Solution:
To solve this problem, we can use the concept of counting techniques and probability.

Total Number of Ways:
Let's first calculate the total number of ways the games can be distributed among the children. Since there are 5 games and 4 children, each game can be given to any of the 4 children. Therefore, the total number of ways is 4^5 = 1024.

Number of Ways with Each Child Getting At Least One Game:
To calculate this, we can use the concept of inclusion-exclusion principle.

Step 1: Each child gets at least one game.
In this case, we can distribute 1 game to each child in 4! ways. After distributing 4 games, we are left with one game which can be given to any of the 4 children. So, the total number of ways in this case is 4! * 4 = 96.

Step 2: Two children get two games each.
In this case, we can choose any two children out of four in 4C2 ways. For each pair of children, we can distribute 2 games to each child in 2! * 2! ways. After distributing the games, we are left with one game which can be given to any of the 4 children. So, the total number of ways in this case is 4C2 * 2! * 2! * 4 = 216.

Step 3: Three children get one game each, and one child gets two games.
In this case, we can choose any three children out of four in 4C3 ways. For each group of three children, we can distribute 1 game to each child in 3! ways. The child who gets two games can be chosen in 4 ways. For the remaining two games, we have only one child left, so there is only 1 way to distribute those games. So, the total number of ways in this case is 4C3 * 3! * 4 * 1 = 96.

Step 4: Four children get one game each, and one child gets one game.
In this case, we can choose any four children out of four in 4C4 ways. For each group of four children, we can distribute 1 game to each child in 4! ways. The child who gets one game can be chosen in 4 ways. So, the total number of ways in this case is 4C4 * 4! * 4 = 96.

Probability:
The probability is given by the number of favorable outcomes divided by the total number of outcomes.

Number of favorable outcomes = Step 1 + Step 2 + Step 3 + Step 4 = 96 + 216 + 96 + 96 = 504
Probability = Number of favorable outcomes / Total number of outcomes = 504 / 1024 = 63 / 128

Simplifying the Probability: