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
2611
In how many ways can the letters of the word 'APPLE' be arranged?
Answer:
60
Step 1: 'APPLE' has 5 letters, so total arrangements without considering repetition = 5! = 120. Step 2: The letter 'P' is repeated 2 times, so we must divide by 2!. Step 3: Total valid arrangements = 5! / 2! = 120 / 2 = 60.
2612
How many arrangements of the word 'HEXAGON' start with 'H' and end with 'N'?
Answer:
120
Step 1: 'HEXAGON' has 7 distinct letters. Fix 'H' at the first position and 'N' at the last. Step 2: The remaining 5 letters (E, X, A, G, O) must be arranged in the middle 5 spots. Step 3: Total ways = 5! = 120.
2613
In how many ways can the letters of the word 'SQUARE' be arranged such that the vowels occupy only odd places?
Answer:
36
Step 1: 'SQUARE' has 6 letters: 3 vowels (U, A, E) and 3 consonants (S, Q, R). Odd positions are 1st, 3rd, 5th. Step 2: Arrange 3 vowels in 3 odd places = 3! = 6 ways. Step 3: Arrange 3 consonants in remaining 3 even places = 3! = 6 ways. Total = 6 × 6 = 36.
2614
In how many ways can the letters of 'ORANGE' be arranged such that the vowels are never together?
Answer:
576
Step 1: Total possible arrangements without restrictions = 6! = 720. Step 2: Number of arrangements where vowels are always together = 144 (from previous question). Step 3: Arrangements where vowels are not together = Total - Vowels together = 720 - 144 = 576.
2615
In how many ways can the letters of the word 'ORANGE' be arranged so that all consonants are always together?
Answer:
144
Step 1: Consonants are R, N, G. Treat them as a single unit (RNG). Step 2: The units are (RNG), O, A, E. Total 4 units = 4! = 24 ways. Step 3: The 3 consonants can arrange internally in 3! = 6 ways. Total = 24 × 6 = 144 ways.
2616
In how many ways can the letters of the word 'ORANGE' be arranged such that the vowels are always together?
Answer:
144
Step 1: The vowels are O, A, E. Treat them as a single unit (OAE). Step 2: The units to arrange are (OAE), R, N, G. Total = 4 units, giving 4! = 24 ways. Step 3: The 3 vowels can arrange among themselves in 3! = 6 ways. Total = 24 × 6 = 144.
2617
How many arrangements of the word 'ORANGE' begin with the letter 'O'?
Answer:
120
Step 1: The letter 'O' is fixed at the first position. Step 2: The remaining 5 letters (R, A, N, G, E) can be arranged in the remaining 5 spots. Step 3: Total ways = 5! = 120.
2618
In how many ways can the letters of the word 'ORANGE' be arranged?
Answer:
720
Step 1: The word 'ORANGE' has 6 distinct letters. Step 2: Number of arrangements is 6!. Step 3: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
2619
How many different 4-letter words can be formed using the letters of the word 'MATH'?
Answer:
24
Step 1: The word 'MATH' has 4 distinct letters. Step 2: Number of arrangements of 4 distinct letters is 4!. Step 3: 4! = 4 × 3 × 2 × 1 = 24.
2620
Which of the following is equivalent to nCr + nC(r-1)?
Answer:
(n+1)Cr
Step 1: This is Pascal's Identity, a fundamental property of combinations. Step 2: It states that adding two adjacent items in Pascal's triangle gives the item directly below them. Step 3: Therefore, nCr + nC(r-1) = (n+1)Cr.