(i) If If ⁵P_r = 2 × ⁶P_r–1, find r.

(ii) If ²⁰P_r = 13 × ²⁰P_r–1, find r.

(iii) If ¹¹P_r = ¹²P_r–1, find r.

(i) To find: the value of r

Formula Used:

Total number of ways in which n objects can be arranged in r places (Such that no object is replaced) is given by,

ⁿP_r

⁵P_r = 2⁶P_r-1

r = 10, 3

Hence, value of r is 3, 10

(ii) To find: the value of r

Formula Used:

Total number of ways in which n objects can be arranged in r places (Such that no object is replaced) is given by,

ⁿP_r

²⁰P_r = 13²⁰P_r-1

21 – r = 13

r = 8

Hence, value of r is 8.

(iii) To find: the value of r

Formula Used:

Total number of ways in which n objects can be arranged in r places (Such that no object is replaced) is given by,

ⁿP_r

¹¹P_r = ¹²P_r-1

(r - 16)(r - 9) = 0

r = 16, 9

Since r cannot be 16 as it creates a negative factorial in denominator. Therefore,
r = 16 is not possible.

Hence, value of r is 9.