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I have this custom function to show the combinations:

x<-function(a,b){

  ncol(combn(a,b))

}

combinations<-numeric()

for(i in 0:12){
    combinations<-c(combinations,x(12,i))
}

combinations

It works, but I want to know if it's possible to do something simpler, like this:

x<-function(a,b){

  ncol(combn(a,b))

}

x(12,0:12)
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The number of combinations of size k from a set of size n can be computed with choose(n, k) in R, and if you have multiple k values you can just pass in a vector for k. Therefore your code can be simplified to:

choose(12, 0:12)
#  [1]   1  12  66 220 495 792 924 792 495 220  66  12   1
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