Efficiently picking combinations of Integers

Let's say we have a 5x5 matrix, filled with 0s.

myMatrix <- matrix(rep(0, 25), ncol = 5)

Now, let's pick a triplet of integers between 1 and 5.

triplet <- c(1,2,3)

For all combinations of this triplet we now add 1 in the matrix, with this function:

addCombinationsToMatrix <- function(.matrix, .triplet){
    indexesToChange <- as.matrix(expand.grid(.triplet, .triplet))
    .matrix[indexesToChange] <- .matrix[indexesToChange] + 1
    .matrix
}

Using the function, we go from

myMatrix

     [,1] [,2] [,3] [,4] [,5]
[1,]    0    0    0    0    0
[2,]    0    0    0    0    0
[3,]    0    0    0    0    0
[4,]    0    0    0    0    0
[5,]    0    0    0    0    0

to

myMatrix <- addCombinationsToMatrix(myMatrix, triplet)
myMatrix

     [,1] [,2] [,3] [,4] [,5]
[1,]    1    1    1    0    0
[2,]    1    1    1    0    0
[3,]    1    1    1    0    0
[4,]    0    0    0    0    0
[5,]    0    0    0    0    0

If we pick another triplet we move on to

nextTriplet <- 2:4
myMatrix <- addCombinationsToMatrix(myMatrix, nextTriplet)
myMatrix

     [,1] [,2] [,3] [,4] [,5]
[1,]    1    1    1    0    0
[2,]    1    2    2    1    0
[3,]    1    2    2    1    0
[4,]    0    1    1    1    0
[5,]    0    0    0    0    0

So, row-column combinations represent how often two integers have been shown together in a triplet: 3 and 4 have been shown together once, 2 and 3 have been shown together twice.

Question: How can one pick triplets, such that every combination (1-2, 1-3, 1-4...) was picked at least once and the number of triplets is minimized.

I'm looking for an algorithm here that picks the next triplet.

Ideally it can be extended to

• arbitrarily big matrices (10x10, 100x100 ...)
• arbitrarily big vectors (quadruplets, quintuplets, n-tuplets)
• an arbitrary number of times a combination must have been picked at least

Example:

myMatrix
myMatrix <- addCombinationsToMatrix(myMatrix, 1:3)
myMatrix
myMatrix <- addCombinationsToMatrix(myMatrix, 3:5)
myMatrix
myMatrix <- addCombinationsToMatrix(myMatrix, c(1,4,5))
myMatrix
myMatrix <- addCombinationsToMatrix(myMatrix, c(2,4,5))
myMatrix