Question 1196471: A team of 6 people is to be chosen from 10 people. Two of the people are sisters who must be separated. Find the number of different teams that can be formed.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Let's say the sisters have codenames of A and B.
For now, let's have sister A be on the team.
That means there are 6-1 = 5 seats left to be filled.
There are 10-2 = 8 people left to fill them.
Sister B cannot be on the team in this case.
There are n = 8 people to pick from, and r = 5 seats to fill
Use the nCr combination formula.
n C r = (n!)/(r!(n-r)!)
8 C 5 = (8!)/(5!*(8-5)!)
8 C 5 = (8!)/(5!*3!)
8 C 5 = (8*7*6*5!)/(5!*3!)
8 C 5 = (8*7*6)/(3!)
8 C 5 = (8*7*6)/(3*2*1)
8 C 5 = (336)/(6)
8 C 5 = 56
There are 56 combinations possible where sister A is on the team, but sister B is not.
Side note: you can use Pascal's Triangle as an alternative to the nCr formula.
The calculations will be identical if sister B was on the team, with sister A off the team.
So we have 56 ways to do this scenario.
That's 56+56 = 112 combinations so far.
Now consider the scenario where both sisters are off the team.
There are n = 10-2 = 8 people to pick from and r = 6 seats to fill.
Apply the nCr combination formula again.
n C r = (n!)/(r!(n-r)!)
8 C 6 = (8!)/(6!*(8-6)!)
8 C 6 = (8!)/(6!*2!)
8 C 6 = (8*7*6!)/(6!*2!)
8 C 6 = (8*7)/(2!)
8 C 6 = (8*7)/(2*1)
8 C 6 = (56)/(2)
8 C 6 = 28
There are 28 ways to have a team where both sisters are off the team.
In total we have 112+28 = 140 different teams
For any given team, the order doesn't matter. This is why we go for nCr instead of nPr.
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Another approach:
This may seem counter-intuitive, but let's have the sisters be on the team together for now.
There are r = 6-2 = 4 seats left to pick with n = 10-2 = 8 people to pick from
Use the nCr combination formula to get 8C4 = 70
There are 70 teams with the two sisters on the team together.
There are 10C6 = 210 possible teams whether we have one sister, both sisters, or no sisters on the team.
That must mean there are 210-70 = 140 teams with at most one sister. Either there's one sister or no sisters on the team.
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Answer: 140
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