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
2571
How many diagonals are there in a regular pentagon?
Answer:
5
Step 1: A pentagon has n=5 vertices. Connecting any two vertices yields either a side or a diagonal. Total lines = 5C2 = 10. Step 2: The number of sides is 5. Step 3: Diagonals = Total lines - Sides = 10 - 5 = 5. (Formula: n(n-3)/2 = 5(2)/2 = 5).
2572
In a tournament, there are 15 teams. Each team plays exactly one match against every other team. What is the total number of matches played?
Answer:
105
Step 1: A match involves 2 teams. Step 2: The number of unique matches is the number of ways to pick 2 teams from 15. Step 3: Matches = 15C2 = (15 × 14) / 2 = 105.
2573
At a party, every person shakes hands with every other person exactly once. If there are 10 people, how many handshakes occur?
Answer:
45
Step 1: A handshake requires exactly 2 people. Step 2: This is a combination problem of selecting 2 people from 10. Step 3: Number of handshakes = 10C2 = (10 × 9) / 2 = 45.
2574
A squad has 15 players. A team of 11 needs to be selected. However, 2 specific players are injured and cannot play. How many ways can the team be selected?
Answer:
78
Step 1: Exclude the 2 injured players. The pool drops to 15 - 2 = 13 players. Step 2: We still need a full team of 11. Step 3: Selection ways = 13C11 = 13C2 = (13 × 12) / 2 = 78.
2575
In how many ways can a cricket team of 11 be chosen out of 12 players if the captain must always be included?
Answer:
11
Step 1: The captain is automatically selected. We need to pick 10 more players. Step 2: The remaining pool of players is 11. Step 3: We need to choose 10 from 11, which is 11C10 = 11C1 = 11 ways.
2576
From a group of 10 people, a team of 3 is selected. If two specific people refuse to be on the team together, how many valid teams can be formed?
Answer:
112
Step 1: Total possible teams = 10C3 = (10×9×8)/6 = 120. Step 2: Number of teams where both specific people ARE together. If both are on the team, we need 1 more person from the remaining 8: 8C1 = 8. Step 3: Valid teams = Total teams - Teams with both = 120 - 8 = 112.
2577
Out of 8 students, 4 are to be selected for a quiz. In how many ways can the selection be made if a specific student is never included?
Answer:
35
Step 1: Since one specific student is excluded, the pool of available students drops to 7. Step 2: We must select 4 students from these 7. Step 3: Number of ways = 7C4 = 7C3 = (7 × 6 × 5) / 6 = 35.
2578
Out of 8 students, 4 are to be selected for a quiz. In how many ways can the selection be made if a specific brilliant student must always be included?
Answer:
35
Step 1: Since one specific student is always included, we only need to select 3 more students. Step 2: The pool of remaining students is 7. Step 3: The number of ways is 7C3 = (7 × 6 × 5) / (3 × 2 × 1) = 35.
2579
In how many ways can a committee of 5 be formed from 7 men and 6 women such that it consists of exactly 3 men and 2 women?
Answer:
525
Step 1: Choose 3 men from 7: 7C3 = (7×6×5)/(3×2×1) = 35. Step 2: Choose 2 women from 6: 6C2 = (6×5)/2 = 15. Step 3: Total ways = 7C3 × 6C2 = 35 × 15 = 525.
2580
From a group of 6 men and 5 women, a committee of 4 is to be formed. How many ways can this be done if it must contain at least 1 woman?
Answer:
315
Step 1: Total ways to form a committee of 4 from 11 people = 11C4 = 330. Step 2: Ways to form a committee with NO women (only men) = 6C4 = 15. Step 3: Ways with at least 1 woman = Total ways - Ways with no women = 330 - 15 = 315.