i am studying for a discrete math final and i do not understand the provided solution. specifically, how to rationalize "c(12,10) = c(12,2)". be as specific and detailed as possible.
how many solutions does the equation have, x1 + x2 + x3 = 10, where x1, x2, and x3 are non-negative integers?
to count the number of solutions, we note that a solution corresponds to the ways of selecting 10 items fro m a set with three elements so that x1 items of type one, x2 items of type two, x3 items of type three. hence, the number of solutions is equal to the number of 10 - combinations with repetitions allowed from a set with three elements. from theorem 2, the equation has c(3 + 10−1, 10) = c(12,10) = c(12,2) = 66 solution