nCr Definitions and Properties

Q. If ${}^n{C}_{r-1}=36,{}^n{C}_r=84and{}^n{C}_{r+1}=126$ then r is

1. 2
2. 3
3. none of these
4. 1

Q. The digit at units place of the sum $(1!{)}^2+(2!{)}^2+(3!{)}^2+.....+(20009{)}^2$

1. 5
2. 0
3. 1
4. 7

Q. If P (10, r) = 5040, then r =

1. 4
2. 6
3. 8
4. 2

Q. A tetrahedron has vertices at $O(0,0,0),A(1,2,1),B(2,1,3)$ and $C(-1,1,2)$. Then the angle between the faces $OAB$ and $ABC$ will be

1. ${120}^{\circ }$
2. ${\cos }^{-1}\left(\frac{17}{31}\right)$
3. ${30}^{\circ }$
4. ${90}^{\circ }$

Q. The value of ${\sum }_{r=1}^{10}r{.}^r{P}_r$ =

1. 11!
2. 11! - 1
3. 11! + 1
4. 11! - 11

Q.

If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}$ = 30800:1, then r =

1. 41

2. 31

3. 51

4. 40

Q. The value of the expression ${}^{47}{C}_4+{\sum }_{j=1}^5{}^{52-j}{C}_3$ is

1. ${}^{47}{C}_5$
2. ${}^{52}{C}_5$
3. ${}^{52}{C}_4$
4. $Noneofthese$

Q. The value of $\underset{n\to \infty }{lim}(3^n+5^n+7^n{)}^{1/n}$ equals

1. $3$
2. $7$
3. $5$
4. $\infty$

Q. Find the number of ways in which $AAABBB$ can be placed in the squares of Fig as shown, so that no row remains empty.

Q.

Find the number of diagonals from one vertex for a polygon having 25 sides.

1. 22
2. 24
3. 20
4. 23

Q. If the letters of the word ASSASSIN are arranged at random in row, the probability that no two S’s occur together is:

1. $\frac{3}{5}$
2. $\frac{1}{28}$
3. $\frac{1}{35}$
4. $\frac{1}{14}$