1. I'm very happy because I can go to school and study now.

2. I'm very happy that I can go to school and study now.

3. I'm very happy I can go to school and study now.
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Are both the same in meaning? Which one is popular, 'because' or 'that'?
Is #3 correct where 'that' is deleted.

All are correct.

The second and third are most commonly used.

All three of the sentences you provided have the same meaning, and they are all grammatically correct. The difference lies in the word choice and the level of formality.

Sentence 1: "I'm very happy because I can go to school and study now."
In this sentence, the word "because" is used as a conjunction to show cause and effect. It explains the reason for your happiness, emphasizing that the ability to go to school and study is the cause of your happiness.

Sentence 2: "I'm very happy that I can go to school and study now."
In this sentence, the word "that" is used as a conjunction to introduce a subordinate clause. It acts as a connector between the main clause ("I'm very happy") and the subordinate clause ("I can go to school and study now").

Sentence 3: "I'm very happy I can go to school and study now."
In this sentence, the word "that" is omitted. This is a common practice in informal speech or writing. Omitting "that" does not change the meaning of the sentence; it simply makes it more concise and less formal.

As for popularity, both "because" and "that" are commonly used in similar contexts. The choice between them depends on personal preference and the specific context in which the sentence is used. If you want to express a stronger cause and effect relationship, using "because" might be more suitable. However, if you prefer a smoother flow of the sentence, using "that" can be a good option.

To summarize, all three sentences are correct and have the same meaning. The choice between "because" and "that" depends on personal preference and the level of formality desired. Sentence 3 is acceptable since omitting "that" is common in informal writing or speech.