Permutations & Combinations PYQs Set 7 (PDF)

Set 7 of 11 · Questions 181-210 of 315

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Permutations & Combinations · Set 7 (Q181-210)
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181. $All the letters of the word ANIMAL are permuted in all possible ways and the permutations thus formed are arranged in dictionary order. If the rank of the word ANIMAL is x, then the permutation with rank x, among the permutation obtained by permuting the letters of the word PERSON and arranging the permutations thus formed in dictionary order is$ (1) $ENOPRS$ (2) $NOSPRE$ (3) $NOEPRS$ (4) $ESORNP$ 182. $The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is$ (1) $16800$ (2) $33600$ (3) $18000$ (4) $14800$ 183. $Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?$ (1) $2!3!4!$ (2) $(3!)^{3} ∙ (4!)$ (3) $(3!)^{2} ∙ (4!)$ (4) $3! (4!)^{3}$ 184. $Let N be the set of positive integers. The number of distinct triplets (x, y, z) satisfying x, y, z \in N, x < y < z and x + y + z = 12 is$ (1) $5$ (2) $7$ (3) $6$ (4) $8$ 185. $The sum of the four digit even numbers that can be formed with the digits 0, 3, 5, 4 with out repetition is$ (1) $14684$ (2) $43536$ (3) $46526$ (4) $52336$ 186. $The number of functions f from 1, 2, 3, \dots, 20 onto 1, 2, 3, \dots, 20 such that f (k) is a multiple of 3, whenever k is a multiple of 4 is:$ (1) $6^{5} \times (15)!$ (2) $5! \times 6!$ (3) $(15)! \times 6!$ (4) $5^{6} \times 15$ 187. $If N (n) = n \prod_{r = 1}^{2023} (n^{2} - r^{2}) (n > 2023), then^{N} C_{N - 1} when n = 2024 is$ (1) $(4047)!$ (2) $(4048)!$ (3) $(6023)!$ (4) $(6069)!$ 188. $If^{2 n} C_{3} :^{n} C_{3} = 12 : 1, then n =$ (1) $5$ (2) $8$ (3) $10$ (4) $3$ 189. $If a polygon of n sides has 560 diagonals then n =$ (1) $35$ (2) $36$ (3) $37$ (4) $38$ 190. $If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is$ (1) $103$ (2) $102$ (3) $101$ (4) $104$ 191. $The number of different words that can be formed from the letters of the word "INTERMEDIATE" such that two vowels never come together, is$ (1) $\frac{6 !}{2 !} \times \frac{7 !}{2 ! 3 !}$ (2) $\frac{5 !}{2 !} \times \frac{6 !}{3 !}$ (3) $6! \times \frac{7 !}{2 ! 3 !}$ (4) $\frac{6 !}{2 !} \times \frac{6 !}{2 ! 3 !}$ 192. $At an election a voter may vote for any number of candidates not exceeding the number to be elected. If 4 candidates are to be elected out of the 12 contested in the election and voter votes for at least one candidate, then the number of ways in which a voter can vote is$ (1) $793$ (2) $298$ (3) $781$ (4) $1585$ 193. $If^{n} C_{12} =^{n} C_{8} then n is equal to$ (1) $12$ (2) $26$ (3) $6$ (4) $20$ 194. $Let S = {1, 2, 3, ......, 9}. For k = 1, 2,......,5, let N_{k} be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N_{1} + N_{2} + N_{3} + N_{4} + N_{5} =$ (1) $210$ (2) $252$ (3) $126$ (4) $125$ 195. $Let α = \frac{( 4 ! ) !}{( 4 ! ) ^{3!}} and β = \frac{( 5 ! ) !}{( 5 ! ) ^{4!}} . Then :$ (1) $α \in N and β \in / N$ (2) $α \in / N and β \in N$ (3) $α \in N and β \in N$ (4) $α \in / N and β \in / N$ 196. $If n = (210)^{2} (360) (143), then the total number of non - trivial factors of n is$ (1) $256$ (2) $872$ (3) $1504$ (4) $1438$ 197. $Given 5 different green toys, 4 different blue toys and 3 different red toys, how many combinations of toys can be chosen taking at least one green and one blue toy?$ (1) $32 \times 16 \times 4$ (2) $31 \times 15 \times 4$ (3) $32 \times 16 \times 8$ (4) $31 \times 15 \times 8$ 198. $Ten chairs are numbered as 1 to 10 . Three women and two men wish to occupy one chair each. First the women choose the chairs marked 1 to 6 , then the men choose the chairs from the remaining. The number of possible ways is$ (1) $^{6} C_{3} \times^{4} P_{2}$ (2) $^{6} C_{3} \times^{4} C_{2}$ (3) $^{6} P_{3} \times^{4} C_{2}$ (4) $^{6} p_{3} \times^{4} p_{2}$ 199. $If^{n} P_{4} = 1680, then n =$ (1) $6$ (2) $12$ (3) $10$ (4) $8$ 200. $Find the number of 3-digit numbers. (including all numbers) which have any one digit is the average of the other two digits.$ 201. $A test consists of 6 multiple choice questions, each having 4 alternative answers in which a candidate answers all six questions such that exactly four of the answers are correct, is _$ 202. $In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is$ 203. $The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is ________$ 204. $How many 5 digit numbers can be made having exactly two identical digit.$ 205. $The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is______.$ 206. $Number of 4-digit numbers that are less than or equal to 3800 and either divisible by 3 or by 11, is equal to ______$ 207. $Three persons enter in a lift at the ground floor. The lift will go upto 10^{th} floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to____.$ 208. $The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to____.$ 209. $If the number of seven-digit numbers, such that the sum of their digits is even, is m \cdot n \cdot 10^{n}; m, n \in 1, 2, 3, \dots, 9, then m + n is equal to ____.$ 210. $An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is \dots$