Eight different letters of an alphabet are given. Words of four letters from these are formed.The number of such words with at least one letter repeated is :

• $\left(\begin{array}{c}8\\ 4\end{array}\right) - {}^8\mathrm{P}_4$

• $8^4 + \left(\begin{array}{c}8\\ 4\end{array}\right)$

• $8^4 - {}^8\mathrm{P}_4$

• $8^4 - \left(\begin{array}{c}8\\ 4\end{array}\right)$

The number of natural numbers less than 1000, in which no two digits are repeated, is :

• 738

• 792

• 837

• 720

9 balls are to be placed in 9 boxes and 5 of the balls cannot fit into 3 small boxes. The number of ways of arranging one ball in each of the boxes is

• 18720

• 18270

• 17280

• 12780

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

• (12, 6)

• (10, 5)

• (9, 4)

• (16, 7)

p points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

• ${\mathrm{p}}^3 + 3{\mathrm{p}}^2$

• $\frac{1}{2}\left({\mathrm{p}}^3 +\hspace{0.17em}\mathrm{p}\right)$

• $\frac{{\mathrm{p}}^2}{2}\left(5\mathrm{p} - 3\right)$

• ${\mathrm{p}}^2\left(4\mathrm{p} - 3\right)$

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

${}_{15}\mathrm{P}_8 = \mathrm{A} + 8 . {}_{14}\mathrm{P}_7 \Rightarrow \mathrm{A} =$

• ${}_{14}\mathrm{P}_6$

• ${}_{14}\mathrm{P}_8$

• ${}_{15}\mathrm{P}_7$

• ${}_{16}\mathrm{P}_9$

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

• 5

• 6

• 7

• 8