Permutations & Combinations PYQs Set 9 (PDF)

Set 9 of 11 · Questions 241-270 of 315

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Permutations & Combinations · Set 9 (Q241-270)
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241. $The number of words that can be formed using the letters of the word V O W E L so that vowels remains always together is$ (1) $60$ (2) $48$ (3) $36$ (4) $45$ 242. $Line L_{1} of slope 2 and line L_{2} of slope \frac{1}{2} intersect at the origin O . In the first quadrant, P_{1}, P_{2}, \dots, P_{12} are 12 points on line L_{1} and Q_{1}, Q_{2}, \dots, Q_{9} are 9 points on line L_{2} . Then the total number of triangles, that can be formed having vertices at three of the 22 points O, P_{1}, P_{2}, \dots, P_{12}, Q_{1}, Q_{2}, \dots, Q_{9}, is:$ (1) $1026$ (2) $1188$ (3) $1134$ (4) $1080$ 243. $The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is:$ (1) $288$ (2) $360$ (3) $306$ (4) $310$ 244. $The number of all four digit numbers which begin with 4 and end with either zero or five is$ (1) $200$ (2) $64$ (3) $256$ (4) $32$ 245. $Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is$ (1) $1120$ (2) $3360$ (3) $1680$ (4) $560$ 246. $The total number of positive integral solutions (x, y, z) such that x y z = 24 is$ (1) $36$ (2) $45$ (3) $24$ (4) $30$ 247. $A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is$ (1) $485$ (2) $468$ (3) $469$ (4) $484$ 248. $The number of 3-digit odd numbers divisible by 3 that can be formed using the digits 1, 2, 3, 4, 5, 6 when repetition is not allowed is$ (1) $18$ (2) $21$ (3) $24$ (4) $36$ 249. $Consider a rectangle ABCD having 5,7,6,9 points in the interior of the line segments AB, CD, BC, DA respectively. Let α be the numberof triangles having these points from different sides as vertices and β be thenumber of quadrilaterals having these points from different sides as vertices. Then (β - α) is equal to :$ (1) $795$ (2) $1173$ (3) $1890$ (4) $717$ 250. $If the sides AB, BC and CA of a triangle ABC have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :$ (1) $364$ (2) $240$ (3) $333$ (4) $360$ 251. $In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:$ (1) $102$ (2) $42$ (3) $1$ (4) $38$ 252. $From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is-$ (1) $atleast 500 but less than 750$ (2) $atleast 750 but less than 1000$ (3) $atleast 1000$ (4) $less than 500$ 253. $Let S be the circle in the xy-plane defined by the equation x^{2} + y^{2} = 4. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve$ (1) $(x + y)^{2} = 3xy$ (2) $x^{2/3} + y^{2/3} = 2^{4/3}$ (3) $x^{2} + y^{2} = 2xy$ (4) $x^{2} + y^{2} = x^{2} y^{2}$ 254. $The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :$ (1) $5880$ (2) $840$ (3) $960$ (4) $5760$ 255. $The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3,5, 7, 9 without repetition, is$ (1) $6$ (2) $12$ (3) $120$ (4) $72$ 256. $The number of five-digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to$ (1) $132$ (2) $120$ (3) $72$ (4) $96$ 257. $The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is .$ 258. $In how many ways 4 square are can be chosen on a chess-board, such that all the squares lie in a diagonal line.$ 259. $Let n \geq 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is$ 260. $The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 is$ 261. $The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is_____$ 262. $The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit “1” and they all are multiple of 11, is .$ 263. $All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is .$ 264. $The number of three-digit even numbers, formed by the digits 0,1,3,4,6,7 if the repetition of digits is not allowed, is .$ 265. $The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is____,$ 266. $Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1,2,3,4,5 and 6 without repetition of digits. Then the total number of such numbers is_____________.$ 267. $The total number of two digit numbers ’ n ’, such that 3^{n} + 7^{n} is a multiple of 10, is$ 268. $The students S_{1}, S_{2}, \dots, S_{10} are to be divided into 3 groups A, B and C such that each group has at least onestudent and the group C has at most 3 students. Then the total number of possibilities of forming such groups is$ 269. $Consider a 20-sided convex polygon K, with vertices A_{1}, A_{2}, . . . . , A_{20} in that order. Find the number of ways in which three sides of K can be chosen so that every pair among them has at least two sides of K between them. (For example (A_{1} A_{2}, A_{4} A_{5}, A_{11} A_{12}) is an admissible triple while (A_{1} A_{2}, A_{4} A_{5}, A_{19} A_{20}) is not).$ 270. $The missing value in the following figure is Use the logic which gives answer in single digit.$