You are creating a multiset of size 3 of the 3 orbs. (multiset just means the same element can be picked multiple times, and the order still doesn’t matter like regular sets.)
The number of multisets of size k from n options is given by the formula ((n,k)) “n multichoose k”. As other commenters have pointed out, using the stars and bars argument this is equivalent to (n+k-1,k) “n+k-1 choose k”.
You are creating a multiset of size 3 of the 3 orbs. (multiset just means the same element can be picked multiple times, and the order still doesn’t matter like regular sets.)
The number of multisets of size k from n options is given by the formula ((n,k)) “n multichoose k”. As other commenters have pointed out, using the stars and bars argument this is equivalent to (n+k-1,k) “n+k-1 choose k”.
So ((3,3)) = (5,3) = 5!/(3!2!) = 10.