Topic: P&C / Counting Subtopic: Arrangements with repeated letters and adjacency restriction \ Difficulty: Medium-Hard Type: MCQ Estimated Time: 3 min tcolorbox[qbox] In how many distinct arrangements of the letters of the word PARALLEL do no two L's stand ne

Topic: P&C / Counting Subtopic: Distribution of identical objects with mixed bounds \ Difficulty: Medium-Hard Type: TITA Estimated Time: 3 min tcolorbox[qbox] Find the number of ways to distribute 18 identical balls into four distinct boxes (A), (B), (C\

Topic: P&C / Counting Subtopic: Integer solutions under linked constraints \ Difficulty: Hard Type: MCQ Estimated Time: 4 min tcolorbox[qbox] How many ordered triples ((x, y, z)) of positive integers satisfy all of the following: itemize[leftmargin=*,topsep

Topic: P&C / Counting Subtopic: Committee formation with multiple representation constraints \ Difficulty: Hard Type: MCQ Estimated Time: 4 min tcolorbox[qbox] A 6-member committee is to be formed from a group of 5 men, 4 women, and 3 children. The committee

Topic: P&C / Counting Subtopic: Digit counting with position-style restriction \ Difficulty: Hard Type: MCQ Estimated Time: 4 min tcolorbox[qbox] How many 5-digit positive integers are there in which all five digits are distinct, no digit is 0, and the digits

Topic: P&C / Counting Subtopic: Position-based arrangement with order constraints \ Difficulty: Hard Type: TITA Estimated Time: 4 min tcolorbox[qbox] The 9 letters of the word EDUCATION are arranged in a row. In how many of these arrangements do the five vowe

Topic: P&C / Counting Subtopic: Digit counting after removing invalid cases \ Difficulty: Hard Type: TITA Estimated Time: 5 min tcolorbox[qbox] How many 4-digit positive integers between 1000 and 9999 (both inclusive) have digit sum equal to 20 and contain at

Topic: P&C / Counting Subtopic: Circular arrangement with position-distance constraint \ Difficulty: Hard Type: MCQ Estimated Time: 4 min tcolorbox[qbox] Eight distinct persons, including two specific persons (P) and (Q), are to be seated around a round t

Topic: P&C / Counting Subtopic: Bounded partitions with structural constraint \ Difficulty: Very Hard Type: TITA Estimated Time: 6 min tcolorbox[qbox] Find the number of sequences of positive integers ((a_1, a_2, a_3, a_4, a_5)) such that: [ a_1 a_2 a_3 a_

Topic: P&C / Counting Subtopic: Arrangement with adjacency, order, and end-position restrictions \ Difficulty: Very Hard Type: MCQ Estimated Time: 6 min tcolorbox[qbox] Six distinct books are arranged in a row on a shelf: three Mathematics books (M_1, M_2, M

Topic: Probability Subtopic: Conditional probability with two-set overlap \ Difficulty: Medium-Hard Type: MCQ Estimated Time: 3 min tcolorbox[qbox] In a particular school, (60%) of the students play cricket, (40%) play football, and (25%) play both s

Topic: Probability Subtopic: Conditional probability with changed sample space \ Difficulty: Medium-Hard Type: TITA Estimated Time: 3 min tcolorbox[qbox] An urn contains 4 red and 5 blue balls. Two balls are drawn one after the other, without replacement. Giv

Topic: Probability Subtopic: Geometric/turn-based probability with dependent events \ Difficulty: Hard Type: MCQ Estimated Time: 4 min tcolorbox[qbox] Three players (A), (B), and (C) take turns rolling a fair six-sided die in the cyclic order (A, B, C,

Topic: Probability Subtopic: Arrangement-based probability with complement counting \ Difficulty: Hard Type: MCQ Estimated Time: 4 min tcolorbox[qbox] All 11 letters of the word MISSISSIPPI are arranged in a row in a uniformly random order (where all distingu

Topic: Probability Subtopic: Conditional probability with binomial structure \ Difficulty: Hard Type: TITA Estimated Time: 4 min tcolorbox[qbox] A fair coin is tossed 6 times in sequence. Given that the total number of heads obtained is strictly greater than

Topic: Probability Subtopic: Probability after careful parity counting \ Difficulty: Hard Type: TITA Estimated Time: 4 min tcolorbox[qbox] Three distinct integers are chosen at random (with all subsets equally likely) from the set (\1, 2, 3, , 15\). The pro

Topic: Probability Subtopic: Probability from random ordering with mixed restrictions \ Difficulty: Hard Type: MCQ Estimated Time: 3 min tcolorbox[qbox] Five guests (A), (B), (C), (D), and (E) line up in a uniformly random order (all (5!) ordering

Topic: Probability Subtopic: Conditional card probability with overlapping categories \ Difficulty: Hard Type: MCQ Estimated Time: 4 min tcolorbox[qbox] Two cards are drawn at random, without replacement, from a standard 52-card deck. In this problem, the fac