# Multisets with 2 Elements

As already mentioned rectangles of size 1 x w can always be constructed by the w=(n+1)*n/2 multisets of cardinality 2 chosen from n items. The 2x3 rectangle isn't possible to get with the 6 multisets of size 2 chosen from n=3 items. The same applies for the 2x5 rectangle using the 10 multisets of size 2 chosen from n=4 items. Otherwise if you take the 6 combinations of 2 elements from n=4 items or the 10 combinations of 2 elements from n=5 items the above rectangles can be made.

## n=5, k=2, 15 multisets

Rectangle 5x3. The solution is very easy to get and it's shown with pieces made from colored stripes.

## n=6, k=2, 21 multisets

Rectangle 7x3. The table tennis balls aren't connected to show that the order of the elements doesn't matter.

## n=7, k=2, 28 multisets

Rectangle 14x2
 5-5 5-6 6-6 3-6 3-3 3-7 7-7 2-7 2-2 2-5 1-5 1-6 4-6 4-5 3-5 5-7 6-7 2-6 2-3 1-3 1-7 4-7 2-4 1-2 1-1 1-4 4-4 3-4

Rectangle 7x4
 6-6 1-6 1-5 1-3 3-3 3-6 2-6 4-6 5-6 5-5 3-5 3-7 2-3 2-2 4-4 4-5 2-5 5-7 7-7 2-7 1-2 3-4 1-4 2-4 4-7 6-7 1-7 1-1

## n=8, k=2, 36 multisets

Rectangle 18x2
 3-3 2-3 2-2 2-7 7-7 4-7 4-4 2-4 2-6 6-6 5-6 5-5 1-5 1-1 1-7 3-7 3-8 8-8 3-4 1-3 1-2 2-8 7-8 5-7 4-5 1-4 4-6 3-6 3-5 2-5 5-8 1-8 1-6 6-7 6-8 4-8

Rectangle 9x4
 4-4 4-5 5-5 3-5 2-3 1-3 1-1 1-6 6-6 3-4 4-8 5-8 2-5 2-2 1-2 1-5 5-6 3-6 3-3 3-8 8-8 2-8 2-4 1-4 1-7 5-7 6-7 3-7 7-8 1-8 6-8 2-6 4-6 4-7 7-7 2-7

Rectangle 6x6. The beads are connected to ensure that all multisets are used.

## n=9, k=2, 45 multisets

Rectangle 9x5
 1-1 1-8 8-8 6-8 6-6 5-6 4-6 3-6 3-8 1-2 1-4 4-8 8-9 6-9 6-7 2-6 1-6 1-3 2-2 2-4 4-4 4-9 1-9 7-9 2-7 1-7 3-7 2-9 2-3 3-4 4-5 1-5 5-9 2-5 5-7 7-7 9-9 3-9 3-3 3-5 5-5 5-8 2-8 7-8 4-7

Rectangle 15x3
 5-5 5-7 7-7 2-7 2-2 2-4 4-4 1-4 4-6 6-6 3-6 6-8 1-8 7-8 6-7 1-5 5-9 7-9 3-7 2-3 2-8 4-8 4-9 4-5 5-6 2-6 6-9 1-6 1-7 4-7 1-1 1-9 9-9 3-9 3-3 3-8 8-8 8-9 5-8 3-5 2-5 2-9 1-2 1-3 3-4

## n=10, k=2, 55 multisets

Rectangle 11x5
 4-4 4-8 8-8 8-10 10-10 7-10 1-7 1-5 5-9 9-9 9-10 2-4 2-8 3-8 3-10 6-10 2-10 1-2 1-8 8-9 6-9 2-9 2-2 2-5 5-8 5-10 4-10 1-10 1-1 1-6 6-8 4-6 4-9 2-6 2-7 7-8 5-7 4-7 1-4 1-9 1-3 3-6 5-6 4-5 6-6 6-7 7-7 7-9 3-7 3-4 3-9 3-3 2-3 3-5 5-5

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