In how many ways can Vasya release his 4 cats outside to roam freely?

In how many ways can Vasya release his 4 cats outside to roam freely? - briefly

Vasya has 4 cats, and each cat can either be released outside or kept indoors. The number of ways Vasya can release his cats outside is determined by the combinations of these choices. The total number of ways is 16, calculated as 2 raised to the power of 4 (2^4).

In how many ways can Vasya release his 4 cats outside to roam freely? - in detail

When considering the number of ways Vasya can release his four cats outside to roam freely, it is essential to understand the permutations and combinations involved. Each cat has two primary states: either inside or outside. Therefore, for each cat, there are two possible choices.

To determine the total number of ways Vasya can decide which cats to release, we need to consider all possible combinations of these states. Since each of the four cats can either be inside or outside, the total number of combinations can be calculated using the formula for combinations with repetition, which is given by (n + k - 1) choose (k - 1), where n is the number of choices for each cat (2 in this case) and k is the number of cats (4 in this case).

However, a simpler approach is to recognize that each cat has 2 options independently of the others. Thus, the total number of ways to decide the state of all four cats is the product of the number of options for each cat. This can be mathematically represented as 2^4.

Calculating this, we get: 2^4 = 16

Therefore, there are 16 different ways Vasya can decide which of his four cats to release outside to roam freely. These combinations include:

1. All four cats inside.
2. All four cats outside.
3. Cat 1 outside, others inside.
4. Cat 2 outside, others inside.
5. Cat 3 outside, others inside.
6. Cat 4 outside, others inside.
7. Cats 1 and 2 outside, others inside.
8. Cats 1 and 3 outside, others inside.
9. Cats 1 and 4 outside, others inside.
10. Cats 2 and 3 outside, others inside.
11. Cats 2 and 4 outside, others inside.
12. Cats 3 and 4 outside, others inside.
13. Cats 1, 2, and 3 outside, Cat 4 inside.
14. Cats 1, 2, and 4 outside, Cat 3 inside.
15. Cats 1, 3, and 4 outside, Cat 2 inside.
16. Cats 2, 3, and 4 outside, Cat 1 inside.

Each of these combinations represents a unique way Vasya can choose to release his cats outside. This detailed breakdown illustrates the various possibilities and ensures a comprehensive understanding of the problem.