ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Permutations, Combinations and Probability (Set 6): 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 distinct arrangements of the letters of BANANA are no two A's adjacent?
Answer: A. 12
Explanation: Arrange B, N, N first: 3!/2! = 3 ways. They leave 4 gaps (including the ends), and the three A's go into 3 different gaps: C(4, 3) = 4 ways. Total 3 × 4 = 12.
Q2. In how many ways can 5 different boys and 3 different girls stand in a row so that no two girls are next to each other?
Answer: B. 14400
Explanation: Arrange the boys in 5! = 120 ways. This creates 6 gaps (including both ends); the 3 girls go into 3 different gaps in P(6, 3) = 6 × 5 × 4 = 120 ways. Total 120 × 120 = 14400.
Q3. How many five-digit numbers divisible by 4 can be made by using each of the digits 1, 2, 3, 4, 5 exactly once?
Answer: C. 24
Explanation: The last two digits must form a multiple of 4: 12, 24, 32 or 52 (4 choices). The other three digits fill the first three places in 3! = 6 ways, so 4 × 6 = 24.
Q4. How many triangles can be formed with vertices among those of a regular decagon such that no side of the triangle is also a side of the decagon?
Answer: B. 50
Explanation: Total triangles C(10, 3) = 120. With exactly two decagon sides: 10. With exactly one decagon side: 10 × 6 = 60 (each side with a third vertex not adjacent to it). So triangles with none = 120 − 10 − 60 = 50.
Q5. In how many distinct arrangements of the letters of MISSISSIPPI do all four S's stay together?
Answer: C. 840
Explanation: MISSISSIPPI has 11 letters: M, IIII, SSSS, PP. Treat SSSS as one block: the 8 objects M, I, I, I, I, P, P and [SSSS] give 8!/(4! 2!) = 840 arrangements.
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Q6. A panel of 4 is chosen from 5 teachers and 4 students so that it contains at least one teacher and at least one student. The number of such panels is:
Q7. In how many ways can 4 men and 4 women sit around a round table so that men and women alternate? (Rotations of the same arrangement count as one.)
Q8. In how many ways can 7 people sit around a circular table so that two particular people are not next to each other?
Q9. If all arrangements of the letters A, H, M, T are written in dictionary order, the rank of MATH is:
Q10. Two fair dice are rolled. Given that at least one shows a 5, the probability that the sum is 8 is:
Q11. Three cards are drawn without replacement from a standard deck. The probability that all three are hearts is:
Q12. Three people are chosen and each is equally likely to have been born in any of the 12 months. The probability that all three were born in different months is:
Q13. Urn A has 3 white and 2 black balls; urn B has 1 white and 4 black balls. An urn is picked at random and a ball drawn is white. The probability that it came from urn A is:
Q14. Events E and F are independent with P(E) = 0.7 and P(F) = 0.2. The probability that at least one of them occurs is:
Q15. If the odds in favour of an event are 3 to 5, its probability is:
Q16. For events with P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2, the probability that neither occurs is:
Q17. If ⁿC₈ = ⁿC₁₂, then ⁿC₁₇ equals:
Q18. If ⁿP₄ = 20 · ⁿP₂, then n equals:
Q19. The number of distinct arrangements of the letters of the word LEVEL is:
Q20. The number of ways of choosing 3 numbers from 1, 2, …, 20 whose sum is even is:
Q21. In how many ways can 3 men and 3 women be seated in a row so that no two men and no two women are adjacent?
Q22. How many 5-digit numbers divisible by 5, with all digits distinct, can be formed using the digits 0 to 9 (no leading zero)?
Q23. A 3-digit number is chosen at random from 100 to 999. The probability that it is divisible by 7 is:
Q24. The digits 1, 2, 3, 4, 5 are arranged at random in a line. The probability that 1 and 2 are next to each other is:
Q25. A fair coin is tossed 4 times. The probability of getting at least 2 heads is:
Q26. A team of 11 is chosen from 15 players. The number of teams in which two particular players are either both included or both excluded is:
Q27. How many rectangles of all sizes (including squares) are there on a 5 × 5 grid of squares?
Q28. How many different 4-digit numbers can be formed using all of the digits 1, 2, 2, 3?
Q29. Three fair dice are rolled. The probability that the sum of the scores is 10 is:
Q30. Two different numbers are chosen at random from 1, 2, …, 10. The probability that their sum is even is: