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Permutation and Combination

1. In how many ways can the letters of the word SPECIAL be arranged in a row such that the vowels occupy only the odd places?

  • 480
  • 576
  • 720
  • 504
Show Explanation
Correct Answer: 576
SPECIAL has 3 vowels (A, E, I) and 4 consonants (S, P, C, L). There are 4 odd positions (1, 3, 5, 7). Vowels can be arranged in odd places in 4P3 ways. = 4 × 3 × 2 = 24 Consonants can be arranged in remaining 4 places in 4! ways. = 24 Total arrangements = 24 × 24 = 576.

2. A man has 12 friends. In how many ways can he invite at least one of them?

  • 4096
  • 4095
  • 2047
  • 2048
Show Explanation
Correct Answer: 4095
Total subsets of 12 friends = 2¹² = 4096 Excluding the case of inviting none: 4096 − 1 = 4095.

3. In how many ways can he invite at least 10 of them?

  • 66
  • 79
  • 140
  • 153
Show Explanation
Correct Answer: 79
Required = ¹²C₁₀ + ¹²C₁₁ + ¹²C₁₂ = 66 + 12 + 1 = 79.

4. In how many ways can the letters of the word VARIOUS be arranged so that vowels and consonants appear alternately?

  • 72
  • 144
  • 288
  • 96
Show Explanation
Correct Answer: 144
VARIOUS has 4 vowels (A, I, O, U) and 3 consonants (V, R, S). Pattern: V C V C V C V Vowels arranged in 4! ways and consonants in 3! ways. = 24 × 6 = 144.

5. How many arrangements can be made by taking 4 letters from the word ADDRESSEE?

  • 9!
  • 370
  • 540
  • 280
Show Explanation
Correct Answer: 370
Considering all selection cases and arranging each case: 5C4×4! + 4C1×4!/3! + 3×4C2×4!/2! + 3×4!/2!2! = 120 + 16 + 216 + 18 = 370.

6. How many four-digit numbers divisible by 3 can be formed using digits 0,1,2,3,8 without repetition?

  • 30
  • 36
  • 42
  • 48
Show Explanation
Correct Answer: 36
Valid digit sets whose sum is divisible by 3: 0123 and 0138 Each gives 18 valid numbers. Total = 18 + 18 = 36.

7. What is the rank of the word RAISE when all arrangements of ARISE are listed alphabetically?

  • 72
  • 75
  • 76
  • 78
Show Explanation
Correct Answer: 76
Words beginning with A, E, I → 4! each = 24 + 24 + 24 Words beginning with RA and next smaller letter combinations → 2! + 1! Total = 24 + 24 + 24 + 2 + 1 + 1 = 76.

8. Find the sum of all possible 5-digit numbers formed using 1,3,4,6,8 without repetition.

  • 4880008
  • 5040068
  • 5866608
  • 5866068
Show Explanation
Correct Answer: 5866608
Formula: (n−1)! × (sum of digits) × 11111 = 4! × (1+3+4+6+8) × 11111 = 24 × 22 × 11111 = 5866608.

9. Among all convex polygons, the maximum number of diagonal intersection points inside is 495. Find n.

  • 10
  • 11
  • 12
  • 13
Show Explanation
Correct Answer: 12
Maximum intersections = nC4 nC4 = 495 Solving gives n = 12.

10. In a regular polygon with 10 sides, how many triangles can be formed having at least one side common with the polygon?

  • 60
  • 65
  • 70
  • 75
Show Explanation
Correct Answer: 70
Number of triangles having at least one side of polygon: = (10C1 × 6C1) + 10 = 60 + 10 = 70.

ONLINE XGMT Course

The Xavier General Management Aptitude Test (XGMT) is the entrance exam for admission to MBA/PGDM programs of Xavier University, Bhubaneswar (XUB). The exam tests a candidate’s aptitude in Quantitative Ability, Verbal Ability, Data Interpretation, Logical Reasoning, and Decision Making.
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