
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
From an urn containing \( N \) balls, of which \( n \) are white and \( N-n \) are black, a selection is made without replacement ( \( n \geq 2 \) ). Consider ordered samples of size \( M \). What is the probability of events \( A_{i, j} \) and \( A_{i, j, k} \), where \( A_{i, j} \) is the event that the \( i \)-th and \( j \)-th positions are occupied by white balls ( \( i < j \leq M \) ), and \( A_{i, j, k} \) is the event that the white balls occupy the positions with numbers \( i, j, k \) ( \( i < j < k \leq M \) )?

Present the answer in LaTex format: \boxed{Your answer}
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