Permutations & Combinations
Counting ordered pairs with remainder condition
nta_pyq_2023_jan
Grade None
Question:
The number of ways of selecting two numbers a and b, \(a \in \{2, 4, 6, \ldots, 100\}\) and \(b \in \{1, 3, 5, \ldots, 99\}\) such that 2 is the remainder when a + b is divided by 23 is
Step-by-Step Solution
Key Concept: Find pairs (a,b) with a$\in$even set, b$\in$odd set such that (a+b) ≡ 2 (mod 23).
a+b=25: 12 pairs; a+b=71: 35 pairs; a+b=117: 42 pairs; a+b=163: 19 pairs. Total = 12+35+42+19 = 108.
Correct Answer: 3