A bag contains 20 identical red balls, 20 identical blue balls, 20 identical green balls, 1 white ball, and 1 black ball. I reach in to get 15 balls. How many outcomes are there? The problem came from a section that teaches multisets so I am required to use $k+n-1\choose k$.
My attempt: $$\text{Case 1: No white and black balls}$$ $$\text{No. of ways = $17\choose 2$}$$ $$\text{Case 2: With white ball only}$$ $$\text{No. of ways = $16\choose 2$}$$ $$\text{Case 3: With black ball only}$$ $$\text{No. of ways = $16\choose 2$}$$ $$\text{Case 4: With white and black}$$ $$\text{No. of ways = $15\choose 2$}$$ Hence, the total number of outcomes is ${17\choose 2}+{16\choose 2}+{16\choose 2}+{15\choose 2}$