Final answer:
To distribute 18 different toys among four children, we calculate the number of ways by raising the number of choices (4) to the power of the number of items (18), resulting in 4^18, which equals 68,719,476,736 different distributions.
Step-by-step explanation:
The task of distributing 18 different toys among four children is a classic problem in combinatorics, a branch of mathematics. To solve this problem, we need to consider each toy as an individual item that can go to any one of the four children. Therefore, for each toy, there are four choices about which child gets it. Considering all 18 toys, we calculate the number of ways to distribute the toys by raising the number of choices to the power of the number of items. This leads us to 4^18 ways to distribute the toys.
This calculation is based on the principle of counting known as the 'Rule of Product' or 'Multiplication Rule', which states that if there are 'n' ways to perform a first action and 'm' ways to perform a second action, then there are n*m ways to perform both actions in sequence. Since the distribution of each toy is an independent event (the distribution of one toy does not affect the distribution of the others), we multiply the choices together for all 18 toys.
Therefore, each of the 18 toys can be given to any of the four children, giving us 4^18, or 68,719,476,736, different possibilities. This number represents the total ways to distribute 18 different toys among four distinct recipients without any restrictions.