Permutations And Combinations

Question
CBSEENMA11014209

Find the number of triangles formed by 12 points, 5 of which are collinear.

Solution

Number of points = 12
Number of traingles formed by these 12 points (no three of which are collinear) by taking 3 at a time
                                             = space space space space space space space straight C presuperscript 12 subscript 3 space equals space fraction numerator 12 factorial over denominator 3 factorial space space 9 factorial end fraction equals fraction numerator 12 cross times 11 cross times 10 over denominator 3 cross times 2 cross times 1 end fraction equals 220.
It is given that 5 points are collinear.
If these 5 points are considered to be distinct such that no three of them are collinear, then the number of triangles formed by these 5 points by taking 3 at a time
                                           equals straight C presuperscript 5 subscript 3 equals fraction numerator 5 factorial over denominator 3 factorial space 2 factorial end fraction equals fraction numerator 5 cross times 4 over denominator 2 cross times 1 end fraction equals 10
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Some More Questions From Permutations and Combinations Chapter

Determine K, so that K + 2, 4K – 6 and 3K – 2 are three consecutive terms of an A.P.