Chapter 7

If in a regular polygon the number of diagonals is 54 , then the number of sides of this polygon is (a) 12 (b) 6 (c) 10 (d) 9

Let \(\mathrm{A}\) and \(\mathrm{B}\) two sets containing 2 elements and 4 elements respectively. The number of subsets of \(\mathrm{A} \times \mathrm{B}\) having 3 or more elements is (a) 256 (b) 220 (c) 219 (d) 211

Let \(\mathrm{T}_{n}\) be the number of all possible triangles formed by joining vertices of an \(n\)-sided regular polygon. If \(\mathrm{T}_{n+1}-\mathrm{T}_{n}=10\), then the value of \(n\) is : (a) 7 (b) 5 (c) 10 (d) 8

On the sides \(\mathrm{AB}, \mathrm{BC}, \mathrm{CA}\) of a \(\triangle \mathrm{ABC}, 3,4,5\) distinct points (excluding vertices \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) ) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are: (a) 210 (b) 205 (c) 215 (d) 220

The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is: \(\quad\) Online April 22, 2013] (a) \({ }^{30} \mathrm{C}_{7}\) (b) \({ }^{21} C_{8}\) (c) \({ }^{21} C_{7}\) (d) \({ }^{30} \mathrm{C}_{8}\)

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is: (a) 40 (b) 41 (c) 16 (d) 32

The number of arrangements that can be formed from the letters \(a, b, c, d, e, f\) taken 3 at a time without repetition and each arrangement containing at least one vowel, is (a) 96 (b) 128 (c) 24 (d) 72

Statement 1: If \(A\) and \(B\) be two sets having \(p\) and \(q\) elements respectively, where \(q>p\). Then the total number of functions from set \(A\) to set \(B\) is \(q^{p}\). Statement 2: The total number of selections of \(p\) different objects out of \(q\) objects is \({ }^{q} \mathrm{C}_{p}\) (a) Statement 1 is true, Statement 2 is false. (b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 . (c) Statement 1 is false, Statement 2 is true (d) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1 .

If the number of 5 -element subsets of the set \(A=\left\\{a_{1}, a_{2}, \ldots, a_{20}\right\\}\) of 20 distinct elements is \(k\) times the number of 5 -element subsets containing \(a_{4}\), then \(k\) is (a) 5 (b) \(\frac{20}{7}\) (c) 4 (d) \(\frac{10}{3}\)

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: (a) \(N \leq 100\) (b) \(100190\)

Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \({ }^{9} C_{3}\). Statement-2: The number of ways of choosing any 3 places from 9 different places is \({ }^{9} C_{3}\). [2011] (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (b) Statement- 1 is true, Statement- 2 is false. (c) Statement- 1 is false, Statement- 2 is true. (d) Statement- 1 is true, Statement- 2 is true; Statement- 2 is a correct explanation for Statement-1.

There are two urns. Urn A has 3 distinct red balls and um B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is [2010] (a) 36 (b) 66 (c) 108 (d) 3

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is: (a) at least 500 but less than 750 (b) at least 750 but less than 1000 (c) at least 1000 (d) less than 500

The set \(S=\\{1,2,3, \ldots \ldots, 12\\}\) is to be partitioned into three sets \(A, B, C\) of equal size. Thus \(A \cup B \cup C=S, A \cap B=B \cap C=A \cap C=\phi\). The number of ways to partition \(\mathrm{S}\) is (a) \(\frac{12 !}{(4 !)^{3}}\) (b) \(\frac{12 !}{(4 !)^{4}}\) (c) \(\frac{12 !}{3 !(4 !)^{3}}\) (d) \(\frac{12 !}{3 !(4 !)^{4}}\)

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is (a) 5040 (b) 6210 (c) 385 (d) 1110

The value of \({ }^{50} C_{4}+\sum_{r=1}^{6}{\underline{\phantom{xx}}}^{56-r} C_{3}\) is (a) \({ }^{55} \mathrm{C}_{4}\) (b) \({ }^{55} C_{3}\) (c) \({ }^{56} \mathrm{C}_{3}\) (d) \({ }^{56} C_{4}\)

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is (a) \({ }^{8} C_{3}\) (b) 21 (c) \(3^{8}\) (d) 5

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 himis (a) 346 (b) 140 (c) 196 (d) 280

If \({ }^{n} C_{r}\) denotes the number of combination of \(n\) things taken \(\mathrm{r}\) at a time, then the expression \({ }^{n} C_{r+1}+{ }^{n} C_{r-1}^{\prime}+2 \times{ }^{n} C_{r}\) equals \(\quad[2003]\) (a) \({ }^{n+1} C_{r+1}\) (b) \({ }^{n+2} C_{r}\) (c) \({ }^{n+2} C_{r+1}\) (d) \({ }^{n+1} C_{r}\)

Five digit number divisible by 3 is formed using \(0,1,2,3,4\), 6 and 7 without repetition. Total number of such numbers are [2002] (a) 312 (b) 3125 (c) 120 (d) 216