I am trying to calculate the probability that a jar with 200 jelly beans and 40 possible jelly bean colors does not feature all colors at least 1 time.
I am aware of the way to solve this using inclusion-exclusion but have been wondering why the following approach does not work.
Let 40 glasses represent the 40 different colors. Now calculate the number of ways in which you can sort jelly beans into these glasses. This is similar to the question of in how many ways we can write 200 as a sum of 40 non-zero integers. The answer to this should be $239\choose 39$.
Now calculate the number of ways to distribute with at least one jelly bean in each glass. Remove 40 jelly beans and add one to each glass. Then we have 160 jelly beans left that we can distribute arbitrarily in any of $199 \choose 39$ ways.
Dividing the latter by the former gives a number close to 0, however, which is clearly wrong. Where am I going wrong with this approach?