Explanation:

The correct answer is option B) since.

In this sentence, the word "since" is used to indicate that the action of the children playing started after the teacher had gone out. It shows a cause-and-effect relationship between the two actions.

Meaning:

The sentence implies that the children started playing after the teacher had left. It suggests that the absence of the teacher allowed the children to begin playing.

Usage:

The word "since" is commonly used to indicate a time relationship between two events. It is used to show that one event happened after another event took place.

Example:

- I haven't seen her since she moved to a different city. (implies that the speaker has not seen the person since the time they moved)
- It has been raining since morning. (implies that the rain started in the morning and is still ongoing)

Other Options:

a) while - This option is incorrect because "while" suggests that two actions are happening at the same time. However, in the given sentence, the children started playing after the teacher had gone out.

c) and - This option is incorrect because "and" is a conjunction used to connect words, phrases, or clauses. It does not convey the cause-and-effect relationship between the two actions in the sentence.

d) or - This option is incorrect because "or" is a conjunction used to present alternatives or choices. It is not appropriate in this sentence as it does not convey the cause-and-effect relationship between the two actions.

Therefore, option B) since is the correct answer as it accurately conveys the cause-and-effect relationship between the children playing and the teacher going out.