We have to choose 4 types of soup from a supermarket's 10 varieties, such that at least 2 are identical. In how many ways can we do this ?
We have 10 varieties of soup like so {v1, .. v10}. We can partition our choices into those with 4 identical (e.g. {v1, v1, v1, v1}), those with 3 identical (e.g. {v1, v1, v1, v2}), and those with 2 identical (e.g. {v1, v1, v2, v3}). Our total number of choices is then:
10 + 10.9 + 10.9C2 = 10 + 90 + 360 = 460
The textbook answer is 505. Can anyone point out my error ?