ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Permutations, Combinations and Probability (Set 7): 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 ordered quadruples (x, y, z, w) of non-negative integers satisfy x + y + z + w = 6?
Answer: A. 84
Explanation: Distribute 6 identical units among 4 variables: C(6 + 4 − 1, 4 − 1) = C(9, 3) = 84.
Q2. The number of distinct arrangements of all the letters of the word ENGINEER is
Answer: C. 3360
Explanation: ENGINEER has 8 letters with E repeated 3 times and N repeated 2 times: 8!/(3!·2!) = 40320/12 = 3360.
Q3. The letters of MATHEMATICS are arranged so that the four vowels A, A, E, I stay together. The number of such arrangements is
Answer: D. 120960
Explanation: Treat the vowel block as one unit: with M, M, T, T, H, C, S there are 8 units, arranged in 8!/(2!·2!) = 10080 ways. Inside the block, the vowels A, A, E, I arrange in 4!/2! = 12 ways. Total = 10080 × 12 = 120960.
Q4. How many four-digit numbers have their digits in strictly increasing order from left to right (for example 1358)?
Answer: D. 126
Explanation: Zero cannot appear (it would have to be the first digit, and then the number is not a four-digit number), so choose any 4 distinct digits from 1–9; there is exactly one increasing arrangement of each choice. C(9, 4) = 126.
Q5. A walker moves from (0, 0) to (4, 4) on a grid, taking unit steps right or up only. The number of such paths that avoid the point (2, 2) is
Answer: C. 34
Explanation: Total paths: C(8, 4) = 70. Paths through (2, 2): C(4, 2)·C(4, 2) = 36. So the paths avoiding it number 70 − 36 = 34.
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Q6. Four letters are to be placed in four addressed envelopes, one letter per envelope. In how many ways can every letter go into a wrong envelope?
Q7. A team of 4 is chosen from 8 people. If two particular people, A and B, refuse to be on the team together, the number of possible teams is
Q8. In how many ways can 9 different people be divided into three unlabelled groups of 3 each?
Q9. Five boys and three girls sit around a circular table so that no two girls are adjacent. The number of arrangements is
Q10. A box holds 20 bulbs of which 4 are defective. Three bulbs are drawn at random together. The probability that at most one is defective is
Q11. Machine M₁ makes 60% of a factory's items, 2% of them defective; machine M₂ makes the other 40%, 5% of them defective. An item is found defective. The probability it came from M₂ is
Q12. A two-digit number is chosen at random from 10 to 99. The probability that the product of its digits is even is
Q13. Six different books are placed at random on a shelf. The probability that two particular books are NOT next to each other is
Q14. Three fair dice are thrown. The probability that the product of the scores is even is
Q15. A fair coin is tossed 7 times. The probability of getting exactly 3 heads is
Q16. Two fair dice are thrown. The probability that the sum of the scores is a multiple of 3 is
Q17. A three-digit number is chosen at random from 100 to 999. The probability that its three digits are all different is
Q18. From 10 members, a committee of 4 is chosen and one of the four is appointed its leader. The number of ways of doing this is
Q19. Four men and three women stand in a row. The number of arrangements in which both end positions are occupied by women is
Q20. The letters of COMBINATION are arranged so that the word begins with C and ends with N. The number of such arrangements is
Q21. The value of ¹⁰C₃ + ¹⁰C₄ + ¹¹C₅ is
Q22. If ⁿPᵣ = 336 and ⁿCᵣ = 56, then n + r equals
Q23. A bag has 3 white and 2 red balls. Two balls are drawn one after another with replacement. The probability that both are of the same colour is
Q24. Two fair dice are thrown and the sum of the scores is found to be 8. The probability that one of the dice shows a 2 is
Q25. A and B are independent events with P(A) = 2/7 and P(B) = 1/3. The probability that exactly one of A, B occurs is
Q26. How many five-digit palindromes (numbers that read the same forwards and backwards, with first digit non-zero) are there?
Q27. In how many ways can 3 seats be chosen from a row of 10 seats so that no two chosen seats are adjacent?
Q28. Four people are chosen at random. Assuming each is equally likely to be born on any day of the week, the probability that at least two share the same birth-day of the week is
Q29. P(A) = 0.3, P(B) = 0.5 and P(A ∩ B) = 0.15. Then P(A ∩ B′) equals
Q30. Three identical red flags, two identical blue flags and one green flag are arranged in a row so that no two red flags are adjacent. The number of arrangements is