Permutations & Combinations PYQs Set 10 (PDF)

Set 10 of 11 · Questions 271-300 of 315

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Permutations & Combinations · Set 10 (Q271-300)
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271. $Let P_{1}, P_{2}, ..., P_{15} be points on a circle. The number of distinct triangles formed by points P_{i}, P_{j}, P_{k} such that i + j + k \neq 15, is :$ (1) $12$ (2) $419$ (3) $443$ (4) $455$ 272. $The letters of the word "COCHIN" are permuted and all the permutations are arranged in an alphabetical order as in an english dictionary. Then number of words that appear before the word COCHIN is$ (1) $360$ (2) $192$ (3) $96$ (4) $48$ 273. $The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is :$ (1) $216$ (2) $192$ (3) $120$ (4) $72$ 274. $If the number of all possible permutations of the letters of the word MATHEMATICS in which the repeated letters are not together is 982 (X), then X =$ (1) $5040$ (2) $14400$ (3) $21600$ (4) $86400$ 275. $A natural number has prime factorization given by n = 2^{x} 3^{y} 5^{z}, where y and z are such that y + z = 5 and y^{- 1} + z^{- 1} = \frac{5}{6}, y > z . Then the number of odd divisors of n, including 1, is :$ (1) $11$ (2) $6 x$ (3) $12$ (4) $6$ 276. $If^{n} P_{r} =^{n} P_{r + 1} and^{n} C_{r} =^{n} C_{r - 1}, then the value of r is equal to:$ (1) $1$ (2) $4$ (3) $2$ (4) $3$ 277. $If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is (6!) k then k is equal to$ (1) $2835$ (2) $5670$ (3) $1890$ (4) $945$ 278. $The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS is$ (1) $179$ (2) $177$ (3) $181$ (4) $175$ 279. $The exponent of 6 in 72! is$ (1) $34$ (2) $70$ (3) $17$ (4) $35$ 280. $There are 5 points P_{1}, P_{2}, P_{3}, P_{4}, P_{5} on the side AB, excluding A and B, of a triangle ABC . Similarly there are 6 points P_{6}, P_{7}, \dots, P_{11} on the side BC and 7 points P_{12}, P_{13}, \dots, P_{18} on the side CA of the triangle. The number of triangles, that can be formed using the points P_{1}, P_{2}, \dots, P_{18} as vertices, is :$ (1) $776$ (2) $796$ (3) $751$ (4) $771$ 281. $In a high school, a committee has to be formed from a group of 6 boys M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6} and 5 girls G_{1}, G_{2}, G_{3}, G_{4}, G_{5} . (i) Let α_{1} be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls. (ii) Let α_{2} be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls. (iii) Let α_{3} be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls. (iv) Let α_{4} be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both M_{1} and G_{1} are NOT in the committee together. LIST-I LIST-II (P) The value of α_{1} is (1) 136 (Q) The value of α_{2} is (2) 189 (R) The value of α_{3} is (3) 192 (S) The value of α_{4} is (4) 200 (5) 381 (6) 461 The correct option is$ (1) $(P) \to (4); (Q) \to (6); (R) \to (2); (S) \to (1)$ (2) $(P) \to (1); (Q) \to (4); (R) \to (2); (S) \to (3)$ (3) $(P) \to (4); (Q) \to (6); (R) \to (5); (S) \to (2)$ (4) $(P) \to (4); (Q) \to (2); (R) \to (3); (S) \to (1)$ 282. $15 lines are concurrent at a point P . A line L not passing through P intersects all the 15 lines and forms triangles with them. Then the number of triangles having L as one of its side is$ (1) $310$ (2) $91$ (3) $182$ (4) $105$ 283. $A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is-$ (1) $140$ (2) $196$ (3) $280$ (4) $346$ 284. $Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:$ (1) $170$ (2) $180$ (3) $210$ (4) $190$ 285. $How many 5 digit telephone numbers can be constructed using the digits 0 to 9 , if each number starts with 67 and no digit appears more than once?$ (1) $335$ (2) $336$ (3) $338$ (4) $337$ 286. $The set S = {1, 2, 3 ..........12} is to be partitioned into three sets A, B, C of equal size. Thus, A \cup B \cup C = S,A \cap B = B \cap C = A \cap C = φ The number of ways to partition S is -$ (1) $12!/3! (4!)3$ (2) $12!/3!(3!)4$ (3) $12!/(4!)3$ (4) $12!/(3!)4$ 287. $How many different words can be formed by jumbling the letters in the word "MISSISSIPPI" in which no two S are adjacent ?$ (1) $8. 6C4 . 7C4$ (2) $6. 7 .8C4$ (3) $6. 8. 7C4$ (4) $7. 6C4 . 8C4$ 288. $Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is :$ (1) $880$ (2) $629$ (3) $630$ (4) $879$ 289. $Find the number of ways of selecting 4 pens and 3 pencils from a packet of 8 pens and 5 pencils,$ (1) $700$ (2) $^{8} P_{4} \times^{5} P_{3}$ (3) $^{8} P_{4} +^{5} P_{3}$ (4) $700 \times^{8} P_{4}$ 290. $Let Tn be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If Tn+1 - Tn = 10, then the value of n is :$ (1) $7$ (2) $5$ (3) $10$ (4) $8$ 291. $The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is_______.$ 292. $A five letter word is to be formed such that the letters appearing in the odd numbered positions are taken from the letters which appear without repetition in the word "MATHEMATICS". Further the letters appearing in the even numbered positions are taken from the letters which appear with repetition in the same word "MATHEMATICS". The number of ways in which the five letter word can be formed is:$ 293. $If^{1} P_{1} + 2 \cdot^{2} P_{2} + 3 \cdot^{3} P_{3} + \dots + 15 \cdot^{15} P_{15} =^{q} P_{r} - s, 0 \leq s \leq 1, then^{q + s} C_{r - s} is equal to ____.$ 294. $The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.$ 295. $The number of ways of giving 20 identical oranges to 3 children such that each child gets at least one orange is_____$ 296. $If^{2 n + 1} P_{n - 1} :^{2 n - 1} P_{n} = 11 : 21, then n^{2} + n + 15 is equal to$ 297. $All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by W_{n} . Let the probability P (W_{n}) of choosing the word W_{n} satisfy P (W_{n}) = 2 P (W_{n - 1}), n > 1. If P (CDBEA) = \frac{2 ^{α}}{2 ^{β} - 1}, α, β \in N, then α + β is equal to: ________.$ 298. $In maths paper there is a question on "Match the column" in which column A contains 6 entries & each entry of column A corresponds to exactly one of the 6 entries given in column B (and vice versa) written randomly. 2 marks are awarded for each correct matching & 1 mark is deducted from each incorrect matching. A student having no subjective knowledge decides to match all the 6 entries randomly. Find the number of ways in which he can answer, to get atleast 25 % marks in this question.$ 299. $Let ABC be a triangle. Consider four points p_{1}, p_{2}, p_{3}, p_{4} on the side AB , five points p_{5}, p_{6}, p_{7}, p_{8}, p_{9} on the side BC, and four points p_{10}, p_{11}, p_{12}, p_{13} on the side AC . None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points p_{1}, p_{2}, \dots, p_{13}, is____.$ 300. $Number of integral solutions to the equation x + y + z = 21, where x \geq 1, y \geq 3, z \geq 4 is equal to _____$