ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Permutations, Combinations and Probability (Set 4): 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 ways can all the letters of the word MISSISSIPPI be arranged?
Answer: A. 34650
Explanation: There are 11 letters with I repeated 4 times, S 4 times and P 2 times: 11!/(4! 4! 2!) = 34650.
Q2. The number of different arrangements of the letters of COMMITTEE is:
Answer: D. 45360
Explanation: There are 9 letters with M, T and E each appearing twice: 9!/(2! 2! 2!) = 45360.
Q3. Five boys and four girls stand in a row so that no two girls are adjacent. The number of arrangements is:
Answer: D. 43200
Explanation: Arrange the 5 boys in 5! = 120 ways, creating 6 gaps. Choose 4 gaps for the girls and arrange them: C(6,4) × 4! = 15 × 24. Total = 120 × 360 = 43200.
Q4. Six people sit around a circular table. The number of arrangements in which two particular people are NOT next to each other is:
Answer: B. 72
Explanation: Total circular arrangements = 5! = 120. With the two together: 2 × 4! = 48. So 120 − 48 = 72.
Q5. How many different necklaces can be made by threading 8 distinct beads on a closed string (a necklace may be flipped over)?
Answer: B. 2520
Explanation: For a necklace that can be turned over the count is (8 − 1)!/2 = 5040/2 = 2520.
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Q6. A committee of 4 is chosen from 6 men and 5 women. The number of committees with at least 2 women is:
Q7. If ⁿP₂ = 90, then n equals:
Q8. If ⁿC₈ = ⁿC₁₂, the value of ⁿC₁₇ is:
Q9. A convex polygon with 15 sides has how many diagonals?
Q10. Ten points lie in a plane, four of which are collinear and no other three are collinear. The number of triangles with vertices at these points is:
Q11. How many even four-digit numbers with all digits different can be formed from the digits 0 to 9?
Q12. The number of ways of distributing 8 identical sweets among 3 children, where a child may get none, is:
Q13. Two fair dice are thrown. The probability that the sum is 9 or more is:
Q14. A fair coin is tossed 4 times. The probability of getting at least one head is:
Q15. If P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7, then the events A and B are:
Q16. A bag holds 5 red and 4 blue balls. Two balls are drawn together. The probability that they have the same colour is:
Q17. Two fair dice are thrown and it is known that at least one shows a 6. The probability that the sum is 8 is:
Q18. Urn I holds 3 red and 2 blue balls; Urn II holds 2 red and 4 blue balls. An urn is chosen at random and one ball is drawn. The probability that it is red is:
Q19. The odds in favour of event A are 3 : 5 and the odds against event B are 2 : 3. If A and B are independent, P(A ∩ B) equals:
Q20. Three letters are placed at random in three addressed envelopes, one letter per envelope. The probability that no letter is in its correct envelope is:
Q21. Three different numbers are chosen at random from 1, 2, …, 10. The probability that all are odd is:
Q22. In how many distinct arrangements of the letters of ARRANGE do the two R's lie next to each other?
Q23. At a meeting every person shakes hands once with every other person, and there are 66 handshakes. The number of people is:
Q24. The number of subsets of a 7-element set that have at least 3 elements is:
Q25. A four-digit PIN uses digits 0–9 and no two consecutive digits are the same (repeats elsewhere are allowed). The number of such PINs is:
Q26. Three fair dice are thrown. The probability that at least one shows a 6 is:
Q27. Events A and B are independent with P(A) = 1/3 and P(B) = 1/4. The probability that exactly one of them occurs is:
Q28. If P(A) = 0.6, P(B) = 0.5 and P(A ∪ B) = 0.8, then P(B | A′) equals:
Q29. From 7 people, a team of 4 is chosen so that two particular people are either both included or both excluded. The number of such teams is:
Q30. Four different prizes are awarded to three students, and a student may receive any number of prizes. The number of ways this can be done is: