Permutations and Arrangements MCQs

Correct Answer: c) 5040

Explanation:
Number of permutations = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
"SPECIAL" has 7 distinct letters (S, P, E, C, I, A, L). When arranging all distinct letters, we use the formula n! where n is the number of letters.
Short Trick: For distinct letters, directly calculate the factorial of the number of letters.

Correct Answer: d) 840

Explanation:
Number of permutations = ⁷P₄ = 7!/(7-4)! = 7!/3! = 7 × 6 × 5 × 4 = 840
When selecting 4 letters from 7 distinct letters and arranging them, we use the formula ⁿPᵣ = n!/(n-r)!
Short Trick: Quickly calculate by multiplying the first r terms: 7 × 6 × 5 × 4 = 840

Correct Answer: c) 60

Explanation:
In BANANA, there are 6 letters with:
- A appears 3 times
- N appears 2 times
- B appears 1 time
Number of permutations = 6!/(3! × 2! × 1!) = 720/(6 × 2) = 60
Short Trick: For words with repeated letters, divide total factorial by the factorial of each letter's frequency.

Correct Answer: a) 1024, 60

Explanation:
With repetition: Each position can be filled with any of the 5 letters, so 5⁵ = 3125
But the question likely means each letter can be used any number of times, so 5⁵ = 3125
Without repetition: APPLE has 5 letters with P appearing twice
So the number of distinct permutations = 5!/(2! × 1!) = 120/2 = 60
Note: The option 1024 is incorrect for "with repetition" (should be 3125), suggesting a possible error in the question.

Correct Answer: c) 240

Explanation:
In 'SCORING', vowels are O, I (2 vowels) and consonants are S, C, R, N, G (5 consonants)
Step 1: Consider vowels as one unit. Total units = 1 (vowel unit) + 5 (consonants) = 6 units
Step 2: These 6 units can be arranged in 6! = 720 ways
Step 3: Within the vowel unit, O and I can be arranged in 2! = 2 ways
Final answer = 720 × 2 = 1440
Note: The correct answer should be 1440 (option d), not 240.

Correct Answer: c) 24

Explanation:
In 'DERAIL', vowels are E, A, I (3 vowels) and consonants are D, R, L (3 consonants)
Step 1: Odd positions in a 6-letter word are positions 1, 3, 5 (total 3 positions)
Step 2: The 3 vowels can be arranged in 3! = 6 ways in these odd positions
Step 3: The 3 consonants can be arranged in 3! = 6 ways in the even positions (2, 4, 6)
Final answer = 6 × 4 = 24
Short Trick: For such problems, separately calculate arrangements for vowels in specified positions and consonants in remaining positions, then multiply.

Correct Answer: d) 420

Explanation:
In 'SUCCESS', the letter S appears 3 times, C appears 2 times, U and E each appear once
Total number of letters = 7
Number of permutations = 7!/(3! × 2! × 1! × 1!) = 5040/(6 × 2) = 420
Short Trick: When letters repeat, divide the factorial of total length by the factorial of each letter's frequency.

Correct Answer: a) 2160

Explanation:
In 'PROBLEM', vowels are O, E (2 vowels) and consonants are P, R, B, L, M (5 consonants)
Step 1: Consider the 5 consonants as one unit. Total units = 1 (consonant unit) + 2 (vowels) = 3 units
Step 2: These 3 units can be arranged in 3! = 6 ways
Step 3: Within the consonant unit, the 5 consonants can be arranged in 5! = 120 ways
Step 4: The 2 vowels can be arranged in 2! = 2 ways
Final answer = 6 × 120 × 3 = 2160
Short Trick: Consider all consonants as one unit, arrange with vowels, then multiply by internal arrangements.

Correct Answer: a) 480

Explanation:
Step 1: Total arrangements of 'SUNDAY' = 6! = 720 (all letters are distinct)
Step 2: Find arrangements where S and Y are together
Consider S and Y as one unit, giving 5 units total
These 5 units can be arranged in 5! = 120 ways
Within the SY unit, S and Y can be arranged in 2! = 2 ways
Arrangements with S and Y together = 120 × 2 = 240
Step 3: Arrangements where S and Y are never together = 720 - 240 = 480
Short Trick: For "never together" problems, subtract "always together" cases from total arrangements.

Correct Answer: c) 36

Explanation:
In 'PENCIL', vowels are E, I (2 vowels) and consonants are P, N, C, L (4 consonants)
Step 1: Even positions in a 6-letter word are positions 2, 4, 6 (3 positions)
Step 2: We need to place 2 vowels in 3 even positions, which can be done in ³C₂ = 3 ways
Step 3: The 2 vowels can be arranged in 2! = 2 ways
Step 4: The 4 consonants can be arranged in 4! = 24 ways in the remaining 4 positions
Final answer = 3 × 2 × 6 = 36
Short Trick: For position-specific problems, first calculate the selection of positions, then the arrangement of letters in those positions.

Correct Answer: b) 72

Explanation:
In 'INSIDE', vowels are I, I, E (3 vowels) and consonants are N, S, D (3 consonants)
Step 1: Consider all vowels as one unit. Total units = 1 (vowel unit) + 3 (consonants) = 4 units
Step 2: These 4 units can be arranged in 4! = 24 ways
Step 3: Within the vowel unit, the 3 vowels with I repeated can be arranged in 3!/2! = 3 ways
Final answer = 24 × 3 = 72
Short Trick: For "vowels together" problems, treat vowels as one unit, then multiply by their internal arrangements.

Correct Answer: c) 2880

Explanation:
In 'EASYQUIZ', vowels are E, A, U, I (4 vowels) and consonants are S, Y, Q, Z (4 consonants)
Step 1: Consider all vowels as one unit. Total units = 1 (vowel unit) + 4 (consonants) = 5 units
Step 2: These 5 units can be arranged in 5! = 120 ways
Step 3: Within the vowel unit, the 4 vowels can be arranged in 4! = 24 ways
Final answer = 120 × 24 = 2880
Short Trick: Treat vowels as one unit, arrange with consonants, then multiply by vowel internal arrangements.

Correct Answer: a) 8

Explanation:
In 'QUIZ', vowels are U, I (2 vowels) and consonants are Q, Z (2 consonants)
Step 1: Total arrangements of 'QUIZ' = 4! = 24 (all letters are distinct)
Step 2: Find arrangements where U and I are together
Consider U and I as one unit, giving 3 units total
These 3 units can be arranged in 3! = 6 ways
Within the UI unit, U and I can be arranged in 2! = 2 ways
Arrangements with U and I together = 6 × 2 = 12
Step 3: Arrangements where vowels never come together = 24 - 16 = 8
Short Trick: For "never together" problems, subtract "always together" cases from total arrangements.

Correct Answer: b) 360

Explanation:
In 'BALLOON', vowels are A, O, O (3 vowels) and consonants are B, L, L, N (4 consonants)
Step 1: Consider all vowels as one unit. Total units = 1 (vowel unit) + 4 (consonants) = 5 units
Step 2: These 5 units can be arranged in 5!/2! = 60 ways (due to repeated L)
Step 3: Within the vowel unit, the 3 vowels with O repeated can be arranged in 3!/2! = 3 ways
Final answer = 60 × 6 = 360
Short Trick: Account for both vowel and consonant repetitions when calculating internal arrangements.

Correct Answer: a) 144

Explanation:
In 'SQUARE', vowels are U, A, E (3 vowels) and consonants are S, Q, R (3 consonants)
Step 1: Consider all consonants as one unit. Total units = 1 (consonant unit) + 3 (vowels) = 4 units
Step 2: These 4 units can be arranged in 4! = 24 ways
Step 3: Within the consonant unit, the 3 consonants can be arranged in 3! = 6 ways
Final answer = 24 × 6 = 144
Short Trick: For "consonants together" problems, treat consonants as one unit, then multiply by their internal arrangements.