UGC NET Questions (Paper – 1)

Each of the k digit positions in a base-r system can take r possible values independently. By the rule of product, the total number of possible combinations is r×r×…×r (k times), which is r^k. Therefore, r^k distinct values can be represented with k digits.

Option A:
Option A correctly uses combinatorial reasoning: each position offers r choices and there are k positions. This yields r^k different patterns, each corresponding to a unique value.

Option B:
Option B, r^k − 1, represents the maximum numeric value but not the count of distinct values. It conflates range endpoint with cardinality.

Option C:
Option C, r^(k−1), counts only patterns with effectively k−1 positions, ignoring the contribution of one place.

Option D:
Option D, r^(k+1), exceeds the actual number of combinations because it assumes additional digit positions that are not present.