How many ways can I write the sum of 10 using 3 elements and the sum of 6 using 2 elements together? With elements from {0,1,2,..}

Question

I want to find how many ways I can write 10 as a sum of 3 elements. Using the stars and bars argument, it would be ${n + r -1 \choose r-1}$ so ${10 + 3 -1 \choose 3-1}$. Likewise, with 6 as a sum of 2 elements is ${6+2-1\choose 2-1}$. But what happens if I want if I want to do both at the same time? i.e. $(x_1,x_2,x_3,x_4,x_5 ): x_1 + x_2 + x_3 = 10$ AND $x_4 + x_5 = 6 $. Using elements from {0,1,2,...}