All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
2561
In how many ways can 6 people sit around a round table so that two specific people DO NOT sit together?
Answer:
72
Step 1: Total ways for 6 people in a circle = (6-1)! = 5! = 120. Step 2: Ways they sit together = 48 (from previous calculation). Step 3: Ways they do not sit together = Total - Together = 120 - 48 = 72.
2562
In how many ways can 6 people sit around a circular table if 2 specific people insist on sitting next to each other?
Answer:
48
Step 1: Treat the 2 specific people as a single unit. Total units = 5. Step 2: Arrange 5 units in a circle: (5 - 1)! = 4! = 24 ways. Step 3: The 2 specific people can swap places internally in 2! = 2 ways. Total = 24 × 2 = 48.
2563
In how many ways can 5 people sit around a circular table?
Answer:
24
Step 1: For a circular arrangement, the relative position matters, not the absolute position. Step 2: The formula for n distinct objects in a circle is (n - 1)!. Step 3: (5 - 1)! = 4! = 24.
2564
If there are 4 horizontal parallel lines and 5 vertical parallel lines, how many parallelograms can be formed by their intersections?
Answer:
60
Step 1: A parallelogram is formed by selecting 2 horizontal lines and 2 vertical lines. Step 2: Ways to select 2 horizontal lines = 4C2 = 6. Ways to select 2 vertical lines = 5C2 = 10. Step 3: Total parallelograms = 4C2 × 5C2 = 6 × 10 = 60.
2565
From 12 points in a plane, exactly 5 are collinear. How many distinct triangles can be formed?
Answer:
210
Step 1: If no points were collinear, total triangles = 12C3 = 220. Step 2: The 5 collinear points cannot form triangles among themselves. Triangles lost = 5C3 = 10. Step 3: Valid triangles = 12C3 - 5C3 = 220 - 10 = 210.
2566
There are 10 points in a plane, no three of which are collinear. How many distinct triangles can be formed?
Answer:
120
Step 1: A triangle requires exactly 3 non-collinear points. Step 2: Number of triangles is the number of ways to choose 3 points from 10. Step 3: 10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 120.
2567
Given 12 points in a plane, exactly 4 of which are collinear. How many distinct straight lines can be drawn through them?
Answer:
61
Step 1: If no points were collinear, lines = 12C2 = 66. Step 2: The 4 collinear points would form 4C2 = 6 lines if non-collinear, but they only form 1 line together. Step 3: Total lines = Total combinations - Lost lines + The 1 collinear line = 66 - 6 + 1 = 61.
2568
There are 10 points on a plane, and no three points are collinear. How many distinct straight lines can be formed by joining these points?
Answer:
45
Step 1: A straight line is formed by connecting any 2 distinct points. Step 2: Number of lines is the number of ways to choose 2 points from 10. Step 3: 10C2 = (10 × 9) / 2 = 45.
2569
How many diagonals does a decagon have?
Answer:
35
Step 1: A decagon has n=10 vertices. Step 2: Use the formula for diagonals: n(n - 3) / 2. Step 3: 10(10 - 3) / 2 = 10(7) / 2 = 70 / 2 = 35.
2570
Find the number of diagonals in an octagon.
Answer:
20
Step 1: An octagon has n=8 vertices. Step 2: Use the diagonals formula: n(n - 3) / 2. Step 3: 8(8 - 3) / 2 = 8(5) / 2 = 40 / 2 = 20.