Free Topic-Wise General Studies MCQs
Conquer advanced numeracy with Permutations and Combinations MCQs. Practice standard UPSC-level probability and arrangement questions with comprehensive, professional explanations.
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Explanation: Use the sum of combinations: for each valid case of w women (w = 2 to 5), multiply C(5,w) by C(7,5โw), then add all cases. C(5,2)C(7,3) + C(5,3)C(7,2) + C(5,4)C(7,1) + C(5,5)C(7,0) = 350 + 210 + 35 + 1 = 596.
Explanation: Treat the 5 vowels as a single super-letter. This gives 5 items to arrange (the vowel block plus 4 consonants), yielding 5! arrangements. The vowels within the block can also be arranged in 5! ways. Multiply these together: 120 ร 120 = 14400.
Explanation: First arrange the 4 consonants, which creates 5 possible gaps (including ends). Place the 5 vowels into these 5 gaps, one per gap, which can be done in 5! ways. Multiply: 4! ร 5!. Therefore, the answer is 2880.
Explanation: For an even number, the units digit must be 2, 4, or 6 (3 choices). After fixing the last digit, choose and arrange the first three digits from the remaining 5 digits: P(5,3). Multiply: 3 ร P(5,3). Therefore, the answer is 180.
Explanation: Apply the multiplication and addition principles. Case 1: units digit is 0; arrange the first three digits from the remaining 5: P(5,3). Case 2: units digit is 5; the thousands digit can be any of {1,2,3,4} (4 choices), then arrange the middle two from the remaining 4: 4 ร P(4,2). Add both cases. Therefore, the answer is 108.
Explanation: In circular arrangements, one position is fixed to break rotational symmetry, leaving (nโ1)! ways to arrange n distinct people. Here, fix one person and arrange the remaining 7: 7!. Therefore, the answer is 5040.
Explanation: Treat the two particular persons as a single block. This gives 7 items to arrange around a circle, which can be done in (7โ1)! = 6! ways. The two persons within the block can be arranged in 2! ways. Multiply: 6! ร 2!. Therefore, the answer is 1440.
Explanation: A necklace is a circular arrangement with no distinction between clockwise and counter-clockwise orientations. First arrange 8 beads in a circle: (8โ1)! = 7!. Then divide by 2 to account for flipping. Result: 7!/2. Therefore, the answer is 2520.
Explanation: This is a derangement problem. For n = 4, the number of derangements is !4 = 4!(1 โ 1/1! + 1/2! โ 1/3! + 1/4!) = 9. The formula for derangements of n items is n! ฮฃ(k=0 to n) (โ1)^k / k!.
Explanation: Since 4 books must include at least one of each of the 3 types, the only possible distribution pattern is (2,1,1). Enumerate the three cases: 2 novels + 1 dictionary + 1 poetry, 1 novel + 2 dictionaries + 1 poetry, and 1 novel + 1 dictionary + 2 poetry. For each case multiply the appropriate combinations and sum the results. Therefore, the answer is 270.
Explanation: To find the rank, count permutations that precede 'RANDOM' in dictionary order. The sorted letters are A, D, M, N, O, R. For the first letter R, all words starting with A, D, M, N, or O precede it: 5 ร 5! = 600. For the third letter N (after RA), words with D or M in that position precede it: 2 ร 3! = 12. For the fifth letter O (after RAND), words with M precede it: 1 ร 1! = 1. Add these and add 1 for the word itself: 600 + 12 + 1 + 1 = 614.
Explanation: A diagonal connects two non-adjacent vertices. The total number of line segments joining any two vertices is C(12,2). Subtract the 12 sides of the polygon. Formula: C(n,2) โ n = n(nโ3)/2. Therefore, the answer is 54.
Explanation: Any three non-collinear vertices determine a unique triangle. Since all vertices of a convex polygon are non-collinear, simply choose 3 vertices from 15: C(15,3). Therefore, the answer is 455.
Explanation: Since A is already included, we need to choose 3 more members from the remaining 8 people (excluding B). This is a direct combination: C(8,3). Therefore, the answer is 56.
Explanation: Each of the 3 positions in the code can be filled by any of the 5 vowels, and repetition is permitted. By the multiplication principle, the total number of codes is 5 ร 5 ร 5 = 5ยณ. Therefore, the answer is 125.
Explanation: Since repetition is not allowed, this is a permutation problem. Choose and arrange 3 vowels from 5: P(5,3) = 5!/(5โ3)! = 5 ร 4 ร 3. Therefore, the answer is 60.
Explanation: The first digit (ten-thousands place) cannot be 0, so there are 9 choices. The remaining 4 digits can be any arrangement of 4 digits chosen from the remaining 9 digits (including 0 now): P(9,4). Multiply: 9 ร P(9,4). Therefore, the answer is 27216.
Explanation: A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of all six available digits is 21. We need to choose 4 digits whose sum is divisible by 3, meaning the 2 digits left out must also sum to a multiple of 3 (specifically 3, 6, or 9). The 5 valid pairs to exclude are {1,2}, {1,5}, {2,4}, {3,6}, and {4,5}. This leaves 5 valid 4-digit subsets. Each subset can be arranged in 4! = 24 ways. Total: 5 ร 24 = 120.
Explanation: To reach (5,4) from (0,0), exactly 5 right moves and 4 up moves are needed, for a total of 9 moves. The problem reduces to choosing which 5 of the 9 moves are right (or equivalently which 4 are up): C(9,5) = C(9,4) = 126.
Explanation: This is a stars-and-bars problem for identical objects into distinct groups with each group getting at least one. Give each child 1 chocolate first, leaving 8 chocolates to distribute freely. The formula is C(nโ1, kโ1) where n=12 and k=4: C(11,3). Therefore, the answer is 165.
Explanation: Use the principle of inclusion-exclusion. The total unrestricted distributions are 3โต. Subtract cases where at least one student gets nothing: C(3,1)ร2โต. Add back cases where two students get nothing (since subtracted twice): C(3,2)ร1โต. Formula: 3โต โ C(3,1)ร2โต + C(3,2)ร1โต = 243 โ 96 + 3 = 150.
Explanation: A rectangle is determined by choosing 2 distinct horizontal lines and 2 distinct vertical lines from the grid boundaries. A 4ร3 grid of squares has 5 horizontal and 4 vertical boundary lines. The count is C(5,2) ร C(4,2). Therefore, the answer is 60.
Explanation: First arrange the 5 men, which can be done in 5! ways. This creates 6 possible gaps (including the two ends) where women can be placed. Choose 4 of these 6 gaps and arrange the 4 women in them: C(6,4) ร 4!. Multiply: 5! ร C(6,4) ร 4!. Therefore, the answer is 43200.
Explanation: Since the three positions are distinct, this is a permutation problem. Select and arrange 3 people from 10: P(10,3) = 10 ร 9 ร 8. Therefore, the answer is 720.
Explanation: There are 4! = 24 such numbers. By symmetry, each digit appears in each place value (units, tens, hundreds, thousands) exactly 24/4 = 6 times. The sum of digits is 10, so each place contributes 6 ร 10 = 60. Total sum = 60 ร (1 + 10 + 100 + 1000) = 60 ร 1111 = 66660.
Explanation: Each handshake involves a unique pair of people. The number of ways to choose 2 people from 15 is C(15,2) = 15ร14/2. Therefore, the answer is 105.
Explanation: Choose 2 vowels from 5: C(5,2). Choose 2 consonants from 21: C(21,2). The 4 selected letters can be arranged in 4! ways. Multiply all three by the multiplication principle. Therefore, the answer is 50400.
Explanation: Apply the multiplication principle. The first digit has 5 choices (1, 3, 5, 7, 9). The second digit has 5 choices (0, 2, 4, 6, 8). The third digit has 4 choices (2, 3, 5, 7). Multiply: 5 ร 5 ร 4. Therefore, the answer is 100.
Explanation: First select the team of 5 from 11 players: C(11,5). Then choose and arrange 2 distinct roles (captain and vice-captain) from the 5 selected players: P(5,2). Multiply by the multiplication principle. Therefore, the answer is 9240.
Explanation: Calculate the total unrestricted arrangements (8!) and subtract the arrangements where the two particular books are together (treat them as one block: 7! ร 2!). Result: 8! โ 7! ร 2. Therefore, the answer is 30240.
Explanation: A number is divisible by 4 if its last two digits form a number divisible by 4. From the digits {1,2,3,4,5,6}, the valid two-digit endings are: 12, 16, 24, 32, 36, 52, 56, 64 (8 possibilities). For each valid ending, the hundreds digit can be any of the remaining 4 digits. Total: 8 ร 4 = 32.
Explanation: Apply Pascal's identity: C(n, k) + C(n, k+1) = C(n+1, k+1). Here n = 15 and k = 3, so C(15,3) + C(15,4) = C(16,4). Therefore, the answer is 1820.
Explanation: Use complementary counting. Total 5-card hands: C(52,5). Subtract hands with no aces (choose all 5 from the 48 non-aces): C(48,5). Result: C(52,5) โ C(48,5). Therefore, the answer is 886656.
Explanation: Treat the three particular persons as a single block. This gives 8 items to arrange around a circle, which can be done in (8โ1)! = 7! ways. The three persons within the block can be arranged in 3! ways. Multiply: 7! ร 3!. Therefore, the answer is 30240.
Explanation: Use the multinomial coefficient for permutations with repeated items. The word has 11 letters with frequencies: M=1, I=4, S=4, P=2. The number of distinct arrangements is 11! / (1! ร 4! ร 4! ร 2!). Therefore, the answer is 34650.
Explanation: Enumerate the valid distributions satisfying both constraints: (2 from A, 4 from B), (3 from A, 3 from B), and (4 from A, 2 from B). For each case multiply the corresponding combinations and sum the results. Therefore, the answer is 805.
Explanation: First find the prime factorization: 360 = 2ยณ ร 3ยฒ ร 5ยน. If a number has prime factorization pโ^a ร pโ^b ร pโ^c, its number of divisors is (a+1)(b+1)(c+1). Here: (3+1)(2+1)(1+1) = 24.
Explanation: Since the balls are identical and boxes are distinct with each box getting at least one, use the restricted stars-and-bars formula: C(nโ1, kโ1) where n = 5 and k = 3. This gives C(4,2) = 6.
Explanation: Enumerate all valid compositions (B, W, A) where B + W + A = 5 and B > W. The valid cases are: (2,0,3), (2,1,2), (3,0,2), (3,1,1), (3,2,0), (4,0,1), (4,1,0), and (5,0,0). For each case, multiply C(6,B) ร C(5,W) ร C(3,A) and sum all the results: 15 + 225 + 60 + 300 + 200 + 45 + 75 + 6 = 926.
Explanation: Each of the 6 distinct seats is distinguishable (front row left, front row middle, etc.). Therefore, this is simply arranging 6 people in 6 distinct positions: 6!. Therefore, the answer is 720.
Explanation: A number is divisible by 5 if its last digit is 0 or 5. Case 1 (ends in 0): arrange the first 4 digits from the remaining 6 digits: P(6,4). Case 2 (ends in 5): the first digit can be any of {1,2,3,4,6} (5 choices, excluding 0 and 5), then arrange the middle 3 from the remaining 5 digits: 5 ร P(5,3). Add both cases. Therefore, the answer is 660.
Explanation: The word has letter frequencies E=3, N=3, G=2, I=2, R=1. Enumerate valid selection patterns for 4 letters: (3,1) gives 8 combinations, (2,2) gives 6, (2,1,1) gives 24, and (1,1,1,1) gives 5. Sum: 8 + 6 + 24 + 5 = 43.
Explanation: Use inclusion-exclusion. Unrestricted distributions: C(10+4โ1, 4โ1) = C(13,3). Subtract cases where at least one child gets 5 or more: C(4,1) ร C(5+4โ1, 3) = 4 ร C(8,3). Add back cases where two children each get 5 or more (since subtracted twice): C(4,2) ร C(0+4โ1, 3) = 6 ร 1. Result: C(13,3) โ 4รC(8,3) + 6 = 286 โ 224 + 6 = 68.
Explanation: There are two possible alternating patterns: starting with a man or starting with a woman. For each pattern, arrange the 5 men in their designated seats (5! ways) and the 5 women in theirs (5! ways). Multiply: 2 ร 5! ร 5!. Therefore, the answer is 28800.
Explanation: Use the gap method or direct transformation. If the chosen seats are xโ < xโ < xโ with gaps of at least 1 between them, define yโ = xโ, yโ = xโ โ 1, yโ = xโ โ 2. Then yโ < yโ < yโ are chosen from 8 positions. The count is C(8,3). Therefore, the answer is 56.
Explanation: For circular arrangements with no two adjacent, use the formula n/(nโk) ร C(nโk, k) or complementary counting. Total ways to choose 3 from 10: C(10,3) = 120. Subtract cases where at least 2 are adjacent: 10 cases of 3 consecutive people, and 10 ร 6 = 60 cases of exactly 2 consecutive. Valid selections: 120 โ 70 = 50.
Explanation: Treat all boys as one block and all girls as another block. The two blocks can be arranged in 2! ways. Within the boys block, arrange 6 boys in 6! ways; within the girls block, arrange 4 girls in 4! ways. Multiply: 2! ร 6! ร 4!. Therefore, the answer is 34560.
Explanation: First choose the chairperson from 12 people: 12 ways. Then choose the remaining 3 committee members from the 11 remaining people: C(11,3). Multiply: 12 ร C(11,3). Alternatively, choose the committee of 4 first (C(12,4) ways) and then choose the chair from the 4 members (4 ways), giving the same result. Therefore, the answer is 1980.
Explanation: Use the multinomial coefficient. The number of ways to partition 6 distinct toys into three groups of 2 is 6! / (2! ร 2! ร 2!) = 90. Alternatively, choose 2 toys for the first child (C(6,2)), 2 for the second (C(4,2)), and the last 2 go to the third child (C(2,2)), then multiply: C(6,2) ร C(4,2) ร C(2,2) = 90.
Explanation: Enumerate valid cases where the number of questions from Section A is 0, 1, 2, or 3 (at most 3). For each case, multiply C(4,A) by C(6,5โA) and sum: C(4,0)C(6,5) + C(4,1)C(6,4) + C(4,2)C(6,3) + C(4,3)C(6,2) = 6 + 60 + 120 + 60 = 246.
Explanation: Of all 7! arrangements, the three particular persons can appear in any of 3! = 6 relative orders, all equally likely. Exactly one of these 6 orders satisfies A before B before C. Therefore, divide the total by 6: 7!/6 = 840.
Explanation: Use complementary counting. Total committees: C(12,6). Subtract invalid cases: committees with 4 women and 2 men (C(5,4)รC(7,2) = 105) and committees with 5 women and 1 man (C(5,5)รC(7,1) = 7). Result: 924 โ 105 โ 7 = 812.
Explanation: Choose the suit first: 4 ways. Then choose 4 cards from the 13 cards of that suit: C(13,4). Multiply by the multiplication principle: 4 ร C(13,4). Therefore, the answer is 2860.
Explanation: A keyring is a circular arrangement with no distinction between clockwise and counter-clockwise orientations. The formula is (nโ1)!/2. For n = 5: 4!/2 = 12.
Explanation: For digits in strictly increasing order, once 3 distinct digits are chosen, there is exactly one way to arrange them in increasing order. Since 0 cannot be the leading digit, choose 3 digits from {1,2,...,9}: C(9,3). Each such choice produces exactly one valid number. Therefore, the answer is 84.
Explanation: First choose 4 couples from the 6 couples: C(6,4). From each of these 4 couples, choose exactly 1 person: 2โด ways. Multiply: C(6,4) ร 2โด = 15 ร 16 = 240. Alternatively, use inclusion-exclusion on C(12,4).
Explanation: The only valid arrangements are strictly alternating: BGBGBGBG or GBGBGBGB. There are 2 patterns. For each pattern, arrange the 4 boys in their seats (4! ways) and the 4 girls in theirs (4! ways). Total: 2 ร 4! ร 4! = 1152.
Explanation: Use complementary counting. Total ways to choose 3 squares from 64: C(64,3). Subtract cases where all 3 are in the same row (8 rows ร C(8,3)) and all 3 are in the same column (8 columns ร C(8,3)). Since no 3 squares can simultaneously be in the same row and same column, there is no overlap to add back. Result: C(64,3) โ 16 ร C(8,3). Therefore, the answer is 40768.
Explanation: First choose which child receives 3 gifts: 3 ways. Then partition the 7 gifts into groups of 3, 2, and 2: C(7,3) ร C(4,2) ร C(2,2) = 35 ร 6 ร 1 = 210. Multiply by 3 for the choice of child: 3 ร 210 = 630.
Explanation: A number is divisible by 9 if the sum of its digits is divisible by 9. By enumerating all 4-element subsets of {1,...,9}, there are exactly 14 subsets whose sum is divisible by 9. Each such subset can be arranged in 4! = 24 ways. Total: 14 ร 24 = 336.