ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Permutations, Combinations and Probability (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a full…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. In how many ways can 5 people sit around a circular table if two particular people must always sit next to each other?
Answer: D. 12
Explanation: Treat the pair as one unit: 4 units around a circle in (4 − 1)! = 6 ways, and the pair can swap in 2 ways: 6 × 2 = 12.
Q2. The number of distinct arrangements of all the letters of the word MISSISSIPPI is:
Answer: B. 34650
Explanation: 11 letters with I ×4, S ×4, P ×2: 11!/(4!·4!·2!) = 34650.
Q3. A and B are independent events with P(A) = 0.6 and P(B) = 0.5. The probability that exactly one of them occurs is:
Answer: B. 0.5
Explanation: P(A only) + P(B only) = 0.6 × 0.5 + 0.4 × 0.5 = 0.3 + 0.2 = 0.5.
Q4. A committee of 4 is to be chosen from 6 boys and 4 girls so that it contains at least 3 girls. The number of ways is:
Answer: C. 25
Explanation: Exactly 3 girls: ⁴C₃ × ⁶C₁ = 24; all 4 girls: ⁴C₄ = 1. Total = 25.
Q5. Two cards are drawn one after the other without replacement from a well-shuffled pack of 52. The probability that both are aces is:
Answer: A. 1/221
Explanation: P = (4/52) × (3/51) = 12/2652 = 1/221.
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Q6. If A and B are mutually exclusive events with P(A) > 0 and P(B) > 0, then:
Q7. The value of ⁶P₄ is:
Q8. A number is chosen at random from 1 to 30. The probability that it is divisible by 3 or by 5 is:
Q9. In how many ways can 4 boys and 3 girls stand in a row so that the 3 girls are always together?
Q10. If the odds against an event are 3 : 5, the probability that the event occurs is:
Q11. Four tosses of a fair coin are made. What is the chance that heads appears on exactly two of them?
Q12. A and B are mutually exclusive events with P(A) = 0.35 and P(B) = 0.4. The probability that neither A nor B occurs is:
Q13. The number of distinct arrangements of the letters of the word BALLOON is:
Q14. A family has two children. Given that at least one of them is a boy, the probability that both are boys is (assume boys and girls equally likely):
Q15. If n!/(n − 2)! = 56, then n equals:
Q16. Using each of the digits 0, 1, 2, 3, 4 at most once, how many four-digit numbers can be written?
Q17. If ⁿP₂ = 30, then n equals:
Q18. In how many ways can the letters of the word FATHER be arranged so that every arrangement begins with F?
Q19. The number of different garlands that can be made from 8 flowers of different colours is:
Q20. A letter is chosen at random from the word PROBABILITY. The probability that it is a vowel (a, e, i, o or u) is:
Q21. In how many ways can 3 different prizes be given to 5 students if a student may receive more than one prize?
Q22. A committee of 5 is to be chosen from 8 people. If one particular person must NOT be included, the number of ways is:
Q23. Two fair dice are thrown. The probability that the sum of the numbers is a prime number is:
Q24. Two fair dice are thrown. The probability of getting a doublet (the same number on both) is:
Q25. For 1 ≤ r ≤ n, the sum ⁿCᵣ + ⁿCᵣ₋₁ is equal to:
Q26. The value of ¹²C₁₀ is:
Q27. Using the digits 1 to 5, each at most once, how many three-digit even numbers can be made?
Q28. Three coins are tossed together. The number of outcomes in the sample space is:
Q29. The number of ways of selecting at least one fruit from 5 different fruits is:
Q30. A polygon has 20 diagonals. The number of its sides is: