Given positive integers and , let be the -colored multiset , where denotes copies of , each with a distinct color. This paper discusses two types of combinatorial identities associated with the permutations and combinations of . The first identity provides, for , an -fold sum for . The second type of identities can be expressed in terms of the Hermite polynomial, and counts color-symmetrical permutations of , which are permutations whose underlying uncolored permutations remain fixed after reflection and a permutation of the uncolored numbers.