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  4. There are 25 points in a plane, of which 10 are on the same line. Of the rest, no three are collinear and no two are collinear with any one of the first ten points. The number of different straight lines that can be formed by joining these points is

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Q. There are $25$ points in a plane, of which $10$ are on the same line. Of the rest, no three are collinear and no two are collinear with any one of the first ten points. The number of different straight lines that can be formed by joining these points is

Permutations and Combinations

Solution:

Out of $25$ given points, $10$ are collinear and hence they form only one straight line. Out of rest of the $15$ points, we have $^{15}C_{2}$ straight lines and any one point out of these $15$ points with any one of $10$ collinear points forms a straight line.
Hence, total straight lines formed
$=\,{}^{15}C_{2} + \,{}^{15}C_{1}\times\,{}^{10}C_{1} +\, 1$
$ = 256$.