Permutations & Combinations
Counting Functions
Grade 11
Question:
<p>Consider all functions \(f : \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}\) which are one-one, onto and satisfy the following property: if \(f(k)\) is odd then \(f(k+1)\) is even, \(k = 1, 2, 3\). The number of such functions is:</p>
<p>(a) 4</p>
<p>(b) 8</p>
<p>(c) 12</p>
<p>(d) 16</p>
Step-by-Step Solution
Key Concept: Use the constraint that whenever \(f(k)\) is odd (values 1 or 3), \(f(k+1)\) must be even (values 2 or 4). Count valid bijections systematically by tracking which elements map to odd vs. even values.
<p>Solution not provided in source text.</p>
Correct Answer: b