there are 10 points in a plane of which 4 are collinear. how many diff. straight lines can be drawn by joining these points.. (please explain the answer by using permutation combination)

Dear ajinkya

straight line can be formed by joining 2 points

so number of ways in which we can select 2 points from 10 points is  = ¹⁰C₂

 but it also include that in which 4 points are in straight line . from those 4 points only one line can be formed .

so we have to subtract ⁴C₂ -1 from above result

so total lines are   = ¹⁰C_{2   -} ⁴C₂ + 1

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Hi, For two points, one line can be drawn. For 3 points, 3 different lines can be drawn, on the condition that these points are not collinear, otherwise only single line will be drawn. Generalizing this, we can conclude that, for n number of points, we can draw a max. of nC2 lines. In this Q., there are 10 points, of which max. of 10C2, or 45 lines can be drawn. However, since 4 of these points are collinear, we have to subtract, the total no. of lines formed by these points, if they were not collinear, as that case is also included in the max. 45 lines formed. So subtracting 4C2 from 10C2 gives you =45-6=39 lines. Here we have subtracted all possible lines formed by those 4 points, however 1 line is formed by those 4 points, since they are collinear, so only 1 line is formed. So finally we have to add 1 line to the total we have got, that is 39. Hence the answer here is, 10C2-4C2+1=45-6+1= 40 lines, is the answer. Thanks

I don`t have an answer I just wanted to ask that why can there be only 1 line formed with 4 points ( sorry for writing it here I didn`t know where to post my doubts )