Answer :

(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,


nPr


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,


nPr


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,


nPr


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


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