In how many ways can A B and C stand in a line with B and C

In how many ways can A, B, and C stand in a line with B and C not next to each other? ANS.:__________________ In how many ways can A, B, C, and D stand in line with B and C not next to each other? ANS.:________________________________ In how many ways can n people stand in a line if two particular ones do not stand next to each other? (Possible extra point for two different methods.) Show steps. How many lines are determined by n points if exactly three of them are collinear? Show steps. How many circles are determined by .10 points if no four of the points all lie on a circle? ANS.:_____________________

Solution

1a.

There can be only 1 way such that B and C are not next to each other and that is when A is between B and C.

Thus the order should be BAC

1b.

There can be 6 ways in which B and C are not next to each other.

BACD, BDCA, BADC, BDAC, CABD, CDBA

1c.

Total number of ways n people can stand in a line =n!

Total number of ways in which n people can stand in line such that two of them are always together = 2*(n-1)!

Thus the required number of ways would be,

N = n!-2*(n-1)! = (n-2)*(n-1)!

1d.

If none of the lines were collinear then the number of lines formed by n points would be C(n,2) where C is the combinatorial function. but since there are 3 points that are collinear we will have to reduce the number of lines not forming because of this collinearity. Usually with 3 points the number of lines formed should be C(3,2) or 3. but since the points are collinear , there will only be a single line which is formed by these 3 points. Thus 2 less lines would be formed from the number of lines it was actually supoosed to form if all points were non-collinear.

Thus the number of lines formed in this case would be,

N = C(n,2) - 2

1e.

Since no four points lie on a circle the number of circles that can be formed with 10 points would be,

N = C(10,3) = 120