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How many triangles can be formed by 15 points, in which 7 of them lie on one line and the remaining 8 on another parallel line? - Mathematics

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प्रश्न

How many triangles can be formed by 15 points, in which 7 of them lie on one line and the remaining 8 on another parallel line?

बेरीज
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उत्तर

To form a triangle we need 3 non-collinear points.

Take the 7 points lying on one line be group A and the remaining 8 points lying on another parallel line be group B.

We have the following possibilities

Group A
7 points
Group B
8 points
Combination
(i) 2 1 7C2 × 8C1
(ii) 1 2 7C1 × 8C2

∴ 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

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Combinations
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 4: Combinatorics and Mathematical Induction - Exercise 4.3 [पृष्ठ १८७]

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सामाचीर कलवी Mathematics - Volume 1 and 2 [English] Class 11 TN Board
पाठ 4 Combinatorics and Mathematical Induction
Exercise 4.3 | Q 23 | पृष्ठ १८७

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