Number of 1 digit numbers = 5
Number of 2 digit numbers = 5² = 25
Number of 3 digit numbers = 5³ = 125

Correct Option: C

Number of 1 digit numbers = 5
Number of 2 digit numbers = 5² = 25
Number of 3 digit numbers = 5³ = 125
Number of 4 digit numbers = 5⁴ = 625
Number of 5 digit numbers = 5⁵ = 3125
∴ Total number of numbers formed with these digits
= 5 + 25 + 125 + 625 + 3125 = 3905

Total number of persons = 9
Host can sit in a particular seat in one way .
Now, remaining positions are defined relative to the host .

Correct Option: B

Total number of persons = 9
Host can sit in a particular seat in one way .
Now, remaining positions are defined relative to the host .
Hence, the remaining can sit in 8 places in ⁸P₈ = 8! ways.
∴ The number of required arrangements = 8! x 1 = 8! = 8! ways

Case I :-
If lady sets on reserved seat, then
2 men can occupy seats from 4 vacant seats in ⁴P₂
= 4 x 3 = 12 ways

Case II :-
If lady does not site on reserved seat, then 1 women can occupy a seat from seat in 4 ways, 1 man can occupy a seat from 3 seats in 3 ways, also 1 man left can occupy a seat from remaining two seats in 2 ways.

Correct Option: B

Case I :-
If lady sets on reserved seat, then
2 men can occupy seats from 4 vacant seats in ⁴P₂
= 4 x 3 = 12 ways

Case II :-
If lady does not site on reserved seat, then 1 women can occupy a seat from seat in 4 ways, 1 man can occupy a seat from 3 seats in 3 ways, also 1 man left can occupy a seat from remaining two seats in 2 ways.

∴ Total ways = 4 x 3 x 2 = 24 ways
Hence, from Case I and case II , total ways = 12 + 24 = 36 ways

first person will shake hands with 11 other persons. Seconds person will shake hands with 10 other persons, Third person will shake hands with 9 other person and so on.

Correct Option: B

first person will shake hands with 11 other persons. Seconds person will shake hands with 10 other persons, Third person will shake hands with 9 other person and so on.
So , total handshake
= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66

OR

2 person are to be chosen from 12 person to have a hand shake
∴ Possible ways = ¹²C₂ = 12! / 2!(12 - 2)! = 66

Neglecting the direction of beads in the chain, number of ways of preparing a chain with 5 different coloured beads
= (1/2) x (5 -1)! = (1/2) x 4! = 24/2 = 12

Correct Option: C

Neglecting the direction of beads in the chain, number of ways of preparing a chain with 5 different coloured beads
= (1/2) x (5 -1)! = (1/2) x 4! = 24/2 = 12