Let's break down the problem systematically and compute the number of tuples generated by each of the queries.

Given: - The table has three attributes ( D1, D2, D3 ) with ( n1, n2, n3 ) distinct values respectively. - The queries use different forms of aggregation: standard GROUP BY, GROUP BY ... WITH CUBE, and GROUP BY ... WITH ROLLUP.

Explanation of Queries

1. Q1: GROUP BY D1, D2, D3:

   • This query simply groups by all combinations of ( D1, D2, D3 ).
   • The total number of combinations = ( n1 \times n2 \times n3 ).

2. Q2: GROUP BY D1, D2, D3 WITH CUBE:

   • The CUBE operation generates aggregates for all possible combinations of the columns, including:
     • Individual values of ( D1, D2, D3 )
     • Combinations of two values (e.g., ( D1, D2 ); ( D1, D3 ); ( D2, D3 ))
     • The overall total (no specific grouping).
   • The total number of combinations for a CUBE with three columns is ( (n1 + 1) \times (n2 + 1) \times (n3 + 1) ).

3. Q3: GROUP BY D1, D2, D3 WITH ROLLUP:

   • The ROLLUP operation generates hierarchical aggregates:
     • Groups by ( D1, D2, D3 )
     • Groups by ( D1, D2 ) only
     • Groups by ( D1 ) only
     • Overall total (no grouping at all).
   • The total number of combinations for a ROLLUP on three columns is ( n1 \times n2 \times n3 + n1 \times n2 + n1 + 1 ).

Applying the Formulas to Identify the Correct Option

From the description, the results should match the formulas: 1. Q1: ( d = n1 \times n2 \times n3 ) 2. Q2: ( e = (n1 + 1) \times (n2 + 1) \times (n3 + 1) ) 3. Q3: ( f = n1 \times n2 \times n3 + n1 \times n2 + n1 + 1 )

Now, let's inspect the given options:

1. Option 1: (2, 2, 2, 6, 18, 8)

   • ( d = 2 \times 2 \times 2 = 8 )
   • ( e = (2 + 1) \times (2 + 1) \times (2 + 1) = 3 \times 3 \times 3 = 27 ) (This does not match 18)
   • ( f = 2 \times 2 \times 2 + 2 \times 2 + 2 + 1 = 8 + 4 + 2 + 1 = 15 ) (This does not match 8)

2. Option 2: (2, 2, 2, 8, 64, 15)

   • ( d = 2 \times 2 \times 2 = 8 )
   • ( e = (2 + 1) \times (2 + 1) \times (2 + 1) = 3 \times 3 \times 3 = 27 ) (This does not match 64)
   • ( f = 2 \times 2 \times 2 + 2 \times 2 + 2 + 1 = 8 + 4 + 2 + 1 = 15 ) (This matches 15)

3. Option 3: (5, 10, 10, 500, 1000, 550)

   • ( d = 5 \times 10 \times 10 = 500 )
   • ( e = (5 + 1) \times (10 + 1) \times (10 + 1) = 6 \times 11 \times 11 = 726 ) (This does not match 1000)
   • ( f = 5 \times 10 \times 10 + 5 \times 10 + 5 + 1 = 500 + 50 + 5 + 1 = 556 ) (This does not match 550)

4. Option 4: (4, 7, 3, 84, 160, 117)

   • ( d = 4 \times 7 \times 3 = 84 )
   • ( e = (4 + 1) \times (7 + 1) \times (3 + 1) = 5 \times 8 \times 4 = 160 ) (This matches 160)
   • ( f = 4 \times 7 \times 3 + 4 \times 7 + 4 + 1 = 84 + 28 + 4 + 1 = 117 ) (This matches 117)

Conclusion

The correct option is:

(4, 7, 3, 84, 160, 117)