(i) If ⁿP₅ = 20 × ⁿP₃, find n.

(ii) If 16 × ⁿP₃ = 13 × ⁿ⁺¹P₃, find n.

(iii) If ²ⁿP₃ = 100 × ⁿP₂, find n.

(i) To find: the value of n

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

ⁿP₅ = 20 × ⁿP₃.

20 = (n - 3)(n - 4)

(n - 8)(n + 1) = 0

n = 8, -1

We know, that n cannot be a negative number.

Hence, value of n is 8

(ii) To find: the value of n

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

16 ×ⁿP₃ = 13 × ⁿ⁺¹P₃.

16n – 32 = 13n + 13

3n = 45

n = 15

Hence, value of n is 15.

(iii) To find: the value of n

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

²ⁿP₃ = 100ⁿP₂

2n(2n - 1)(2n - 2) = 100 × n(n - 1)

4n(2n - 1)(n - 1) = 100 × n(n-1)

8n(n - 13) = 0

n = 0, 13

We know that n should be greater than zero.

Hence, value of n is 13