How many triangles can be formed by joining 15 points when 7 of them are on the same straight line?
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∴ Required number of ways of forming the triangle = (7C2 × 8C1) + (7C1 × 8C2) = `(7!)/(2!(7 - 2)!) xx 8 + 7 xx (8!)/(2!(8 - 2)!)` = `(7!)/(2 xx 5!) xx 8 + 7 xx (8!)/(2! xx 6!)` = `(7 xx 6 xx 5! xx 8)/(2! xx 5!) + (7 xx 8 xx 7 xx 6!)/(2! xx 6!)` = `(7 xx 6 xx 8)/(2! xx 5!) + (7 xx 8 xx xx 6!)/(2! xx 6!)` = `(7 xx 6 xx8)/(2 xx 1) + (7 xx 8 xx 7)/(2 xx 1)` = 7 × 6 × 4 + 7 × 4 × 7 = 168 + 196 = 364 |
