ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Permutations, Combinations and Probability (Set 5): 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. How many 4-digit numbers greater than 4000 can be formed from the digits 1, 2, 3, 4, 5, 6 if no digit is repeated?
Answer: D. 180
Explanation: The first digit is 4, 5 or 6 (3 ways); the other three places are filled in 5 × 4 × 3 = 60 ways, so 3 × 60 = 180.
Q2. In how many ways can the 8 distinct letters of EQUATION be arranged so that the four even places (2nd, 4th, 6th and 8th) are all occupied by vowels?
Answer: B. 2880
Explanation: Choose and arrange 4 of the 5 vowels in the 4 even places: P(5, 4) = 120. The remaining 4 letters fill the other four places in 4! = 24 ways. Total 120 × 24 = 2880.
Q3. How many triangles can be formed from the vertices of a regular octagon if no side of the triangle may be a side of the octagon?
Answer: D. 16
Explanation: Total triangles C(8, 3) = 56. Those with no two vertices adjacent: choose 3 non-adjacent vertices on a circle, (8/5)·C(5, 3) = 16.
Q4. At a gathering every two persons shake hands exactly once, and there are 66 handshakes in all. The number of persons is:
Answer: B. 12
Explanation: n(n − 1)/2 = 66 gives n² − n − 132 = 0, so n = 12.
Q5. The number of ways of choosing 3 cards of the same suit from a standard 52-card deck is:
Answer: A. 1144
Explanation: Choose the suit in 4 ways and 3 cards of that suit in C(13, 3) = 286 ways: 4 × 286 = 1144.
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Q6. From 12 people a committee of 5 is chosen. If two particular people refuse to serve together, the number of possible committees is:
Q7. In how many ways can 6 persons sit around a circular table if two particular persons must not sit next to each other?
Q8. Four married couples sit around a circular table with each husband next to his wife. The number of distinct arrangements is:
Q9. How many 3-digit even numbers can be formed from the digits 0, 1, 2, 3, 4, 5 without repeating a digit?
Q10. The number of distinct arrangements of the letters of COMMITTEE is:
Q11. If all the letters of RAIN are arranged in alphabetical dictionary order, the position of RAIN is:
Q12. If ⁿ⁺¹C₃ = 2 · ⁿC₂, then n equals:
Q13. If ⁿC₈ = ⁿC₁₂, then ⁿC₁₈ equals:
Q14. The sum ¹⁰C₀ + ¹⁰C₂ + ¹⁰C₄ + … + ¹⁰C₁₀ equals:
Q15. The value of ⁸C₃ + ⁸C₄ is:
Q16. Two fair dice are thrown. The probability that the larger of the two numbers shown is exactly 4 is:
Q17. Four fair coins are tossed once. The probability of getting exactly two heads is:
Q18. Two cards are drawn together from a deck of 52. The probability that exactly one of them is an ace is:
Q19. Two fair dice are thrown. Given that both show odd numbers, the probability that their sum is 8 is:
Q20. An urn has 3 white, 4 black and 5 green balls. Three are drawn together. The probability that all three colours appear is:
Q21. A three-digit number is chosen at random. The probability that it is divisible by 7 is:
Q22. A and B are independent events with P(A) = 0.3 and P(B) = 0.6. Then P(A given that B does not occur) equals:
Q23. The odds in favour of an event are 3 : 5. The probability that the event does not occur is:
Q24. If P(A) = 1/3, P(B) = 1/4 and P(A ∩ B) = 1/12, then P(A′ ∩ B′) equals:
Q25. If P(A ∩ B) = 0.12 and P(B) = 0.4, then P(A | B) equals:
Q26. A team of 5 is chosen from 6 boys and 4 girls. The number of teams containing at least 2 girls is:
Q27. In how many ways can the letters of SUCCESS be arranged so that no two S's are adjacent?
Q28. How many 4-letter strings over the letters A, B, C, D, E (repetition allowed) contain at least one repeated letter?
Q29. In how many ways can 3 distinct persons occupy 3 of 6 chairs placed in a row?
Q30. The number of rectangles of all sizes in a 4 × 4 grid of unit squares is: