The value of n, when nP₂ = 20 is _______.

(a)

3

(b)

6

(c)

5

(d)

4

The number of diagonals in a polygon of n seates is equal to _______.

(a)

nC₂

(b)

nC₂ - 2

(c)

nC₂ - n

(d)

nC₂ - 1

The greatest positive integer which divide n(n + 1) (n + 2) (n + 3) for n \(\in\) N is ________.

(a)

2

(b)

6

(c)

20

(d)

24

For all n > 0, nC₁ + nC₂ + nC₃ + ... +nC_n is equal to _______

(a)

2n

(b)

2ⁿ- 1

(c)

n²

(d)

n² - 1

The middle term in the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 10 }\) is _______.

(a)

10C₄\(\left( \frac { 1 }{ x } \right) \)

(b)

10C₅

(c)

10C₆

(d)

10C₇x⁴

Evaluate the following using binomial theorem:(999)⁵

How many triangles can be formed by joining the vertices of a hexagon?

How many distinct words can be formed using all the letters of the following words.
MISSISSIPPI

Find the middle terms in the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 11 }\)

Find x if \(\frac { 1 }{ 6! } +\frac { 1 }{ 7! } =\frac { x }{ 8! } \).

Find the number of arrangements that can be made out of the letters of the word "ASSASSINATION".

Prove that the term independent of x in the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 2n }is\quad \frac { 1.3.5.....,(2n-1){ 2 }^{ n } }{ n! } \)

By the principle of mathematical induction, prove the following.
1³ + 2³ + 3³ + ....... + n³ = \(\frac { { n }^{ 2 }(n+1)^{ 2 } }{ 4 } \) for all \(n\in N\).