Re: There are four distinct pairs of brothers and sisters. In how many way
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16 Dec 2007, 22:28
i'm not even sure of the answer. maybe walker can help out.
4 pairs = 4*2 = 8 people total
8C4 = 8!/4!4! = 70 total outcomes
Total – unfavorable = favorable
Unfavorable outcomes
Assuming one pair of twins in the committee, we have two spaces left. Since we plugged a pair of twins in the committee, we have 8-2= 6 people to fill 2 spaces.
6C2 = 6!/2!4! = 15 ways to fill the two remaining slots
We only filled the slots with one pair, and we have to account for arrangements of the pairs. Now, we have 4 pairs. 4*15= 60 total arrangements
When we place members into the remaining slots, there may be an additional set of twins. There are 2 remaining slots to which we can fit a pair of twins. If it were one remaining slot, we cannot fit a pair of twins, so we wouldn’t have to account for duplicates.
Now we account for the number of duplicates.
# of duplicates = Total arrangements - # of unique combinations
To find the # of duplicates of twins, we need treat a pair of twins as one unit.
This means the 4 slots are really 2 slots.
Total ways of arranging four pairs of twins in two slots
4P2 = 4*3 = 12 total ways
Total # of unique combinations
Choosing two pairs out of 4 pairs
= 4C2
= 6
Therefore, # of duplicates = 12 - 6 = 6 duplicates
70 – 60 + 6 = 16