math for design question

Question

How many ways can 6 pegs be arranged in 12 holes. All 6 pegs must be used each time.

All holes and pegs are identical.

Answer 1

If only one peg is permitted per hole, then there is only one distinguishable way, namely all 6 balls, one per hole.

If multiple pegs are permitted in each hole, then you are asking how many ways can I add the numbers from 0 to 6 so that the sum of the numbers is 6:

6,0

5,1

4,2

3,3

4,1,1

3,2,1

2,2,2

3,1,1,1

2,2,1,1

1,1,1,1,2