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
2551
In how many ways can 5 distinct balls be distributed into 3 distinct boxes, with no restriction on the number of balls in a box?
Answer:
243
Step 1: Each of the 5 balls can be placed in any of the 3 boxes. Step 2: The first ball has 3 choices, the second has 3 choices, and so on. Step 3: Total ways = 3 × 3 × 3 × 3 × 3 = 3⁵ = 243.
2552
Find the number of ways to place 3 distinct letters into 3 addressed envelopes so that none goes to its correct envelope.
Answer:
2
Step 1: Use the derangement formula for n=3. Step 2: D3 = 3!(1/2! - 1/3!) = 6(1/2 - 1/6). Step 3: 6(3/6 - 1/6) = 6(2/6) = 2.
2553
What is the number of ways to put 4 letters into 4 addressed envelopes such that no letter goes into its correct envelope (a complete derangement)?
Answer:
9
Step 1: The formula for derangement Dn is n! [1 - 1/1! + 1/2! - 1/3! + ... + (-1)ⁿ/n!]. Step 2: D4 = 4! (1/2 - 1/6 + 1/24) = 24(1/2 - 1/6 + 1/24). Step 3: 24(12/24 - 4/24 + 1/24) = 24(9/24) = 9.
2554
In how many ways can 6 distinct keys be arranged on a key ring?
Answer:
60
Step 1: A key ring can be flipped over, making clockwise and anti-clockwise arrangements identical. Step 2: Formula is (n - 1)! / 2. Step 3: (6 - 1)! / 2 = 5! / 2 = 120 / 2 = 60.
2555
In how many ways can 4 married couples sit around a circular table if every husband and wife sit together?
Answer:
96
Step 1: Treat each couple as a unit. 4 units arranged in a circle = (4-1)! = 3! = 6 ways. Step 2: Within each unit, the husband and wife can swap places (2! ways each). Since there are 4 couples, internal arrangements = 2⁴ = 16. Step 3: Total ways = 6 × 16 = 96.
2556
In how many ways can 8 people be seated around a circular table if A and B must sit exactly opposite each other?
Answer:
720
Step 1: Fix person A. B must sit opposite to A. This takes 1 way. Step 2: There are 6 remaining seats for the 6 remaining people. Step 3: The 6 people can arrange themselves in these fixed 6 seats in 6! ways. 6! = 720.
2557
In how many ways can a president and 4 committee members sit around a circular table?
Answer:
24
Step 1: There are 5 people in total. Step 2: For a regular circular arrangement of n distinct people, the number of ways is (n - 1)!. Step 3: (5 - 1)! = 4! = 24 ways.
2558
How many garlands can be formed using 10 different flowers?
Answer:
181440
Step 1: A garland can be flipped over, making clockwise and anti-clockwise arrangements identical. Step 2: Formula is (n - 1)! / 2. Step 3: (10 - 1)! / 2 = 9! / 2 = 362880 / 2 = 181440.
2559
How many different necklaces can be made using 7 distinct beads?
Answer:
360
Step 1: For a necklace, turning it over does not create a new arrangement (clockwise and anti-clockwise are identical). Step 2: The formula is (n - 1)! / 2. Step 3: (7 - 1)! / 2 = 6! / 2 = 720 / 2 = 360.
2560
In how many ways can 5 men and 5 women sit around a circular table such that no two men sit together?
Answer:
2880
Step 1: First, seat the 5 women in a circle: (5-1)! = 4! = 24 ways. Step 2: This creates 5 gaps between the women. Step 3: Seat the 5 men in these 5 gaps in a linear fashion: 5! = 120 ways. Total = 24 × 120 = 2880.