## Sprites and witches

### Task number: 3454

How many ways there to arrange \(m\) water sprites and \(n\) witches in a row such that no two sprites are adjacent?

#### Variant

Let \(m=5, n=7\).

Assume that the sprites are indistinguishable, as are the witches.

#### Variant

\( m = 5, n = 7 \), we arrange characters on a line, but they are all recognizable from each other.

#### Variant

\( m = 5, n = 7 \), water sprites, resp. witches are indistinguishable from each other, but we arrange them on a circle under the same condition that no two sprites are adjacent.

#### Variant

\( m = 5, n = 10 \), sprites as well as witches are indistinguishable from each other, and we put them on a circle.

#### Variant

\( m = 6, n = 12 \), sprites as well as witches are indistinguishable from each other, and we put them on a circle.