The hundred place will be reserved for 3 or 2
5 digits are free to fill rest two places i,e., of tens and unit.

Correct Option: A

The hundred place will be reserved for 3 or 2
5 digits are free to fill rest two places i,e., of tens and unit.

Number of required 3 digit number = 2 x ⁵C₂ = 20

We know that, ⁿP_r = ⁿC_rr!
∵ ⁿP_r = 720ⁿC_r
⇒ ⁿC_r.r! = 720ⁿC_r

Correct Option: A

We know that, ⁿP_r = ⁿC_rr!
∵ ⁿP_r = 720ⁿC_r
⇒ ⁿC_r.r! = 720ⁿC_r
⇒ r! = 720
∴ r = 6

Let there be n persons in the room. The total number of hand shakes is same as the number of ways of selecting 2 out of n.
ⁿC₂ = 66

Correct Option: B

Let there be n persons in the room. The total number of hand shakes is same as the number of ways of selecting 2 out of n.
ⁿC₂ = 66
⇒ n(n - 1) / 2! = 66
⇒ n² - n - 132 = 0
⇒ (n - 12) (n + 11) = 0
∴ n = 12

³⁵C_{n + 7} = ³⁵C_{4n - 2}
We know that if ⁿC_x = ⁿC_y then, Either x = y or x + y = n

Correct Option: B

³⁵C_{n + 7} = ³⁵C_{4n - 2}
We know that if ⁿC_x = ⁿC_y then, Either x = y or x + y = n

Case 1: x + y = n
⇒ n + 7 + 4n - 2 = 35
⇒ 5n + 5 = 35
⇒ 5n = 30
∴ n = 6

Case 2: x = y
⇒ n + 7 = 4n - 2
⇒ 4n - n = 2 + 7
⇒ 3n = 9
∴ n = 3

ⁿC_{r - 1} = 36, ⁿC_r = 84 and ⁿC_{r + 1} = 126
⇒ ⁿC_r / ⁿC_{r - 1} = 84/36
⇒ 3n - 10r + 3 = 0 ....(i)

ⁿC_{r + 1} / ⁿC_r = 126/84
⇒ 2n - 5r - 3 = 0 .............(ii)

Correct Option: B

ⁿC_{r - 1} = 36, ⁿC_r = 84 and ⁿC_{r + 1} = 126
⇒ ⁿC_r / ⁿC_{r - 1} = 84/36
⇒ 3n - 10r + 3 = 0 ....(i)

ⁿC_{r + 1} / ⁿC_r = 126/84
⇒ 2n - 5r - 3 = 0 .............(ii)

From (i) and (ii) r = 3, n = 9