The number of different ways of preparing a garland using 6 distinct white roses and 5 distinct red roses such that no two red roses come together is

• 21600

• 43200

• 86400

• 151200

Box I contains 30 cards numbered I to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :

• $\frac{2}{3}$

• $\frac{2}{5}$

• $\frac{8}{17}$

• $\frac{4}{17}$

Let E^c denote the complement of an event E. Let E₁, E₂ and E₃ be any pairwise independent events withP(E₁) > 0 and $\mathrm{P}\left({\mathrm{E}}_1 \cap {\mathrm{E}}_2\cap {\mathrm{E}}_3\right) = 0 \mathrm{then} \left({{\mathrm{E}}_3}^{\mathrm{c}} \cap \raisebox{1ex}{${{\mathrm{E}}_3}^{\mathrm{c}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{E}}_1$}\right.\right)$ is equal to

• $\mathrm{P}\left({{\mathrm{E}}_3}^{\mathrm{c}}\right) - \mathrm{P}\left({\mathrm{E}}_2\right)$

• $\mathrm{P}\left({{\mathrm{E}}_3}^{\mathrm{c}}\right) - \mathrm{P}\left({{\mathrm{E}}_2}^{\mathrm{c}}\right)$

• $\mathrm{P}\left({\mathrm{E}}_3\right) - \mathrm{P}\left({\mathrm{E}}^{\mathrm{c}}\right)$

• $\mathrm{P}\left({{\mathrm{E}}_2}^{\mathrm{C}}\right) + \mathrm{P}\left({\mathrm{E}}_3\right)$

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 ?

• 2! 3! 4!

• 3!(4!)³

• 3! . 2 . (4!)

• (3!)³ . (4!)

If  the coefficient of x is in expansion of ${\left({\mathrm{x}}^2 + \frac{\mathrm{k}}{\mathrm{x}}\right)}^5$ is 270, then k is equal to

• 1

• 2

• 3

• 4

3 out of 6 vertices of a regular hexagon are chosen at a time at random. The probability that the triangle formed with these three vertices is an equilateral triangle, is

• $\frac{1}{2}$

• $\frac{1}{5}$

• $\frac{1}{10}$

• $\frac{1}{20}$

If 12 identical balls are to be placed in 3 identical boxes, then the probability that  one of the boxes contains exactly 3 balls, is

118.

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

• 880

• 629

• 630

• 630

Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ⁹C₃
Statement-2: The number of ways of choosing any 3 places from 9 different places is ⁹C₃ .

• Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

• Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. 

• Statement-1 is true, Statement-2 is false.

• Statement-1 is true, Statement-2 is false.

If C and D are two events such that C ⊂ D and P(D) ≠0, then the correct statement among the following is:

• P(C|D) = P(C)

• P(C|D)  ≥ P(C)

• P(C|D) < P(C)

• P(C|D) < P(C)