Sets, Relations & Functions
General
Grade 11
Question:
The number of functions f from \(\{1, 2, 3, \ldots, 20\}\) onto \(\{1, 2, 3, \ldots, 20\}\) such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :-
15)! \times 6!
56 \times 15
5! \times 6!
6 \times (15)!
Step-by-Step Solution
<div class="solution"><p>f(x)=(2^{1+x}+2^{1-x}+3^x+3^{-x})/2. By AM-GM: 2^{1+x}+2^{1-x}≥4 and 3^x+3^{-x}≥2. Min f=3 at x=0.</p><div class="trap-box"><strong>Trap:</strong> Don't minimize by calculus before seeing the AM-GM form.</div><div class="key-concept"><strong>Key Concept:</strong> aˣ+a⁻ˣ≥2 (AM-GM)</div></div>
Correct Answer: 1