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PERMUTATIONS AND COMBINATIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSPERMUTATIONS AND COMBINATIONS1 Mark
MCQ
Choose the correct answer. The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is.
  • A
    105
  • B
    15
  • C
    175
  • ✓
    185
✓

Answer

Correct option: D.
185
  1. 185
Solution:
Total number of triangle formed from 12 points taking 3 at a time $={ }^{12} \mathrm{C}_3$
But given that out of 12 points 7 are collinear. So, these seven points will form no triangle.
$\therefore$ The required number of triangles $={ }^{12} \mathrm{C}_3-{ }^7 \mathrm{C}_3$
$=\frac{12!}{3!\ 9!}-\frac{7!}{3!\ 4!}=\frac{12\times11\times10\times9}{3\times2\times1\times9!}-\frac{7\times6\times5\times4!}{3\times2\times1\times4!}$
$=\frac{12\times11\times10}{3\times2}-\frac{7\times6\times5}{3\times2}=220-35=185$

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