Visual SolutionPYQ 2024 · Apr 6 Shift 1Standard

Question

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

(A)

48

(B)

56

(C)

24

(D)

16

Solution Path

Total

= {}^{8}C_{3} = 56

. With at least 1 octagon side:

32 + 8 = 40

. No octagon side

= 56 - 40 = 16

01Question Setup

1/4

Regular octagon with vertices labeled

1

8

. Count triangles with NO side coinciding with an octagon side.

Count triangles with no octagon side

02Count Cases

2/4

Total triangles:

{}^{8}C_{3} = 56

. Triangles with exactly 1 octagon side:

8 \times 4 = 32

. Triangles with exactly 2 octagon sides (3 consecutive):

8

{}^{8}C_{3} = 56

, with 1 side:

32

, with 2 sides:

8

03SubtractKEY INSIGHT

3/4

Triangles with at least 1 octagon side

= 32 + 8 = 40

. Subtract from total:

56 - 40 = 16

No octagon side

= 56 - 40 = 16

04Final Answer

4/4

Complementary counting: total minus those with at least one octagon side.

\boxed{16}

- Answer (D)

Concepts from this question1 concepts unlocked

★

STANDARD

Count what you don't want and subtract from total: desired = total - undesired

|A| = |U| - |A^c|

When the constraint is complex, counting the complement is often far simpler. Saves 3-4 minutes on hard P&C problems.

At-least-one problemsDerangementsRestriction problems

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