We throw a die a number of times. We let Y k be the number...
We throw a die a number of times. We let be the number of different faces that came up after throws. Now we want .
According to the answer sheet, we use the principle of inclusion-exclusion here to arrive at the following answer:
(which I believe to be equal to)
But I just can't wrap my head around how the principle of inclusion-exclusion was exactly used here. My first thought was that it has something to do with the fact that if we see 5 faces, we've also seen at least 4 faces, etc. so that is a subset of . I've tried to manually calculate (using the principle of inclusion-exclusion) but I don't arrive at the same answer. So far I haven't come to a good intuitive understanding yet, it would be great if someone could help me grasp this.
Answer & Explanation
Let be the event that face number does not appear, for each . Then event is the same as the event that at least one occurs, i.e. that at least one face is missing. Therefore, using the principle of inclusion exclusion, and the symmetry of the problem,
This is where the equality you were given comes from.
No, you are misinterpreting. The equation you have is
where is the set of faces that show up, and is a subset of faces in each sum.
Expanding a little more, we have 6 events "only 1,2,3,4,5 are allowed", "only 1,2,3,4,6 are allowed", ..., "only 2,3,4,5,6 are allowed", or equivalently, the six events are "1 does not show up", "2 does not show up", ..., "6 does not show up". is the union of these events, and intersecting of these events is precisely reducing the allowable faces to 6-. So inclusion-exclusion gives
which is equation (1).