Permutations and Combinations

$\mathrm{If} {}^{\mathrm{n}}\mathrm{P}_{\mathrm{r}} = 30240 \mathrm{and} {}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}} = 252, \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} \left(\mathrm{n}, \mathrm{r}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• (12, 6)

• (10, 5)

• (9, 4)

• (16, 7)

A polygon has 54 diagonals. Then, the number of its sides is

• 7

• 9

• 10

• 12

A bag contains n white and n black balls. Pairs of balls are drawn at random without replacement successively, until the bag is empty. If the number of ways in which each pair consists of one white and one black ballis 14400, then n is equal to

• 6

• 5

• 4

• 3

The number of five digit numbers divisible by 5 that can be formed using the numbers 0, 1, 2, 3, 4, 5 without repetition is

• 240

• 216

• 120

• 96

$\mathrm{If} {}^{\left(\mathrm{n} - 1\right)}\mathrm{C}_1 + {}^{\left(\mathrm{n} - 1\right)}\mathrm{C}_4 > {}^{\mathrm{n}}\mathrm{C}_3, \mathrm{then} \mathrm{the} \mathrm{minimum} \mathrm{value} \mathrm{of} \mathrm{n} \mathrm{is}$

• 5

• 6

• 7

• 8

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\left({\mathrm{x}}^2 - 1\right)}^7, \mathrm{then} {\mathrm{f}}^{\left(14\right)}\left(\mathrm{x}\right) = ?$

• 0

• 2!

• 7!

• 14!

10 men and 6 women are to be seated in a row so that no two women sit together. The number of ways they can be seated, is

• 11! 10!

• $\frac{11!}{6! 5!}$

• $\frac{10! 9!}{5!}$

• $\frac{11! 10!}{5!}$

If t_n denotes the number of triangles formed with n points in a plane, no three of which are collinear and if t_{n + 1} - t_n= 36_, then n is equal to

• 7

• 8

• 9

• 10

$$\begin{gathered} \mathrm{The} \mathrm{term} \mathrm{independent} \mathrm{of} \mathrm{x}\left(\mathrm{x} > 0, \mathrm{x} \ne 1\right) \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \\ {\left[\frac{\left(\mathrm{x} + 1\right)}{{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - {\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} + 1} - \frac{\left(\mathrm{x} - 1\right)}{\left(\mathrm{x} - \sqrt{\mathrm{x}}\right)}\right]}^{10} \mathrm{is} \end{gathered}$$

• 105

• 210

• 315

• 420

Out of thirty points in a plane, eight of them are collinear. The number of straight lines that can be formed by joining these points, is

• 296

• 540

• 408

• 348