Permutations And Combinations

Step 1: Understanding the Concept:
This question requires calculation and solving of permutations ( ) and combinations ( ).
Step 2: Key Formula or Approach:
Permutation formula:
Combination formula:
Step 3: Detailed Explanation:
(A) :


Value = . So, (A) matches (III).
(B) :
. So, (B) matches (IV).
(C) , find n:
. We need to find four consecutive integers whose product is 360.
We can test values. Let's try n=6: . This is correct.
So, n = 6. (C) matches (I).
(D) , find n:

. We need two consecutive integers whose product is 420.
We know . Let's try n=21.
. This is correct.
So, n = 21. (D) matches (II).
The correct matching is: (A)-(III), (B)-(IV), (C)-(I), (D)-(II).
Step 4: Final Answer:
The correct option that reflects this matching is (3).