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Question: Answered & Verified by Expert
Assertion (A): The number of triangles that can be formed by joining the mid-points of any three adjacent faces of a cube is 20 .

Reason (R): If there are $\mathrm{n}$ points on a plane and none of them are collinear, then the number of triangles that can be formed is $\mathrm{C}(\mathrm{n}, 3)$.
MathematicsPermutation CombinationNDANDA 2006 (Phase 2)
Options:
  • A Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
  • B Both $\mathbf{A}$ and $\mathbf{R}$ are individually true but $\mathbf{R}$ is not the correct explanation of $\mathbf{A}$.
  • C $\mathrm{A}$ is true but $\mathbf{R}$ is false.
  • D $\mathrm{A}$ is false but $\mathbf{R}$ is true.
Solution:
1048 Upvotes Verified Answer
The correct answer is: Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
Number of faces in a cube $=6$ Number of triangles formed by joining mid points of
faces is selection of three points from 6 points $={ }^{6} \mathrm{C}_{3}$
$=\frac{6 !}{3 ! 3 !}=20$
Hence, both $\mathrm{A}$ and $\mathrm{R}$ are individually true and $\mathrm{R}$ is correct explanation of A.

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