- In how many ways can 8 blood samples be divided into 2 groups to be sent to different laboratories for testing if there are 4 samples in each group? Assume that the laboratories are distinguishable.
Picking 4 samples from 8 for one of the groups can be done in \(\binom{8}{4} = \frac{8 \cdot 7 \cdot 6 \cdot 5}{4 \cdot 3 \cdot 2 \cdot 1} = 70\) ways. The remaining 4 samples also automatically constitute the second group. For example if we pick \(\{1, 2, 3, 4\}\) for the first group, then the second group automatically gets the 4 remaining samples, in this case: \(\{5, 6, 7, 8\}\). Later, when we pick \(\{5, 6, 7, 8\}\) for the first group, the second group will get \(\{1, 2, 3, 4\}\). So, we do not need to multiply by 2 as \(\binom{8}{4}\) already accounts for each possible group of 4 samples going to both laboratories.
- In how many ways can 8 blood samples be divided into 2 groups to be sent to different laboratories for testing if there are 4 samples in each group? Assume that the laboratories are indistinguishable.
This can be done in \(\frac{\binom{8}{4}}{2} = 35\) ways (half the count of case a))
- In how many ways can the 8 samples be divided into 2 groups if there is at least 1 item in each group? Assume that the laboratories are distinguishable.
There are \(2^{8} - 2 = 254\) subsets of 8 elements, excluding an empty set and a set with all 8 elements. Similarly to case a) above, we do not need to multiply by 2.