If Vasya has four cats at home, how many ways are there to let them out for a walk? - briefly
There are 16 possible ways to let four cats out for a walk. Each cat can either be let out or not, resulting in 2^4 = 16 combinations.
If Vasya has four cats at home, how many ways are there to let them out for a walk? - in detail
When considering the problem of determining the number of ways to let four cats out for a walk, it is essential to understand the combinatorial principles involved. Each cat can either be let out or not, creating a binary decision for each feline. This scenario can be analyzed through the lens of combinatorics, specifically focusing on the number of subsets of a set.
Firstly, recognize that the set of cats consists of four distinct elements. The total number of subsets of a set with n elements is given by 2^n. This formula arises because each element can either be included in a subset or excluded from it, leading to two choices per element. For four cats, the number of subsets is 2^4, which equals 16. These subsets include the empty set (where no cats are let out) and the set containing all four cats.
To break it down:
- There is 1 way to let out no cats (the empty set).
- There are 4 ways to let out exactly one cat (choosing any one of the four cats).
- There are 6 ways to let out exactly two cats (choosing any two out of the four cats, which is calculated using the binomial coefficient C(4,2)).
- There are 4 ways to let out exactly three cats (choosing any three out of the four cats, which is calculated using the binomial coefficient C(4,3)).
- There is 1 way to let out all four cats (the set containing all four cats).
Summing these possibilities, we get: 1 (no cats) + 4 (one cat) + 6 (two cats) + 4 (three cats) + 1 (all cats) = 16 ways.
Thus, there are 16 different ways to decide which cats to let out for a walk. This comprehensive analysis illustrates the power of combinatorial mathematics in solving problems involving discrete choices.