Permutations & Combinations PYQs Set 6 (PDF)

Set 6 of 11 · Questions 151-180 of 315

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Permutations & Combinations · Set 6 (Q151-180)
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151. $Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to$ (1) $18$ (2) $16$ (3) $12$ (4) $15$ 152. $Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is$ (1) $5^{6}$ (2) $\frac{1}{2} \cdot 6!$ (3) $6!$ (4) $\frac{5}{2} \cdot 6!$ 153. $The number of even numbers greater than 1000000 that can be formed using all the digits 1, 2, 0, 2, 4, 2 and 4 is$ (1) $120$ (2) $240$ (3) $310$ (4) $480$ 154. $The number of triplets (x, y, z) where x, y, z are distinct non negative integers satisfying x + y + z = 15, is$ (1) $80$ (2) $136$ (3) $114$ (4) $92$ 155. $Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is :-$ (1) $199$ (2) $201$ (3) $101$ (4) $200$ 156. $From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is$ (1) $145$ (2) $165$ (3) $155$ (4) $135$ 157. $The least value of n for which^{(n - 1)} C_{2} +^{(n - 1)} C_{3} >^{n} C_{2} is$ (1) $7$ (2) $4$ (3) $5$ (4) $6$ 158. $What is the rank of the word "MOTHER" when all possible words are formed using all its letters and arranged as in a dictionary?$ (1) $308$ (2) $309$ (3) $291$ (4) $307$ 159. $There are 10 points in a plane, out of these 6 are collinear. If N is the number of triangles formed by joining these points. then :$ (1) $N \leq 100$ (2) $100 < N \leq 140$ (3) $140 < N \leq 190$ (4) $N > 190$ 160. $The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is$ (1) $1579$ (2) $1581$ (3) $1578$ (4) $1580$ 161. $In a plane there are 37 straight lines of which 13 pass through point A and 11 pass through the point B . Moreover, no three lines (apart from the lines passing through A and B) pass through same point and no two are parallel. What is the number of points of intersection of the straight lines?$ (1) $^{37} C_{2}$ (2) $^{37} C_{2} -^{13} C_{2} -^{11} C_{2}$ (3) $^{37} C_{2} -^{13} C_{2} -^{11} C_{2} + 2$ (4) $^{37} C_{2} - 2$ 162. $If^{n} C_{r} denotes the number of combinations of n distinct things taken r at a time, then the domain of the function g (x) =^{(16 - x)} C_{(2 x - 1)} is$ (1) ${1, 2, 3, 4, 5}$ (2) ${0, 1, 2, 3, 4}$ (3) $ϕ$ (4) ${0}$ 163. $The least value of n such that^{(n - 1)} C_{6} +^{(n - 1)} C_{7} <^{n} C_{8} is$ (1) $14$ (2) $15$ (3) $16$ (4) $17$ 164. $If^{9} C_{3} +^{9} C_{5} =^{10} C_{r} for some r \in N, then r =$ (1) $3$ (2) $4$ (3) $5$ (4) $7$ 165. $How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?$ (1) $120$ (2) $21600$ (3) $720$ (4) $3600$ 166. $If n, r are two positive integers such that 1 \leq r < n, then^{n} P_{r + 1} + r^{2}^{n - 1} P_{r - 1} + (r + 1)^{n - 1} P_{r} + r^{n - 1} P_{r - 1} =$ (1) $^{n + 2} P_{r + 2}$ (2) $^{n + 2} P_{r + 1}$ (3) $(n + 1)!$ (4) $^{n + 1} P_{r + 1}$ 167. $If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440^{th} position in this arrangement, is:$ (1) $PRNAUK$ (2) $PRKANU$ (3) $PRKAUN$ (4) $PRNAKU$ 168. $The number of strictly increasing functions f from the set 1, 2, 3, 4, 5, 6 to the set 1, 2, 3, \dots, 9 such that f (i) \neq = i for 1 \leq i \leq 6, is equal to :$ (1) $27$ (2) $22$ (3) $21$ (4) $28$ 169. $The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is$ (1) $21$ (2) $20$ (3) $22$ (4) $18$ 170. $If the letters of the word REGULATIONS be arranged in such a way that relative positions of the letters of the word GULATIONS remain the same, then the probability that there are exactly 4 letters between R and E is$ (1) $\frac{3}{55}$ (2) $\frac{6}{55}$ (3) $\frac{9}{55}$ (4) $\frac{7}{55}$ 171. $The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is____.$ 172. $The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is_______.$ 173. $Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A \subseteq S : A \neq φ and the sum of all the elements of A is not a multiple of 3} is .$ 174. $Let b1b2b3b4 be a 4-element permutation with {1, 2, 3, ........., 100} for 1i4 and for, such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1b2b3b4 is equal to________.$ 175. $Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is _____$ 176. $Find the number of positive integers less than 2310 which are relatively prime with 2310.$ 177. $Six cards are drawn one by one from a set of unlimited number of cards, each card is marked with numbers -1, 0 or 1. Number of different ways in which they can be drawn if the sum of the numbers shown by them vanishes, is:$ 178. $Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is________.$ 179. $Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to _______$ 180. $The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _________$