Sets, Relations & Functions
General
Grade 11
Question:
<p>If \(|X|=5\), \(|Y|=7\), \(\alpha=\)#injections \(X\to Y\), \(\beta=\)#surjections \(Y\to X\). Find \(\frac{\beta-\alpha}{5!}\).</p>
Step-by-Step Solution
<div class="solution"><p>\alpha=^7P_5=2520. \beta=5^7-C(5,1)4^7+C(5,2)3^7-C(5,3)2^7+C(5,4)·1=16800. (16800-2520)/120=119.</p><p><strong>Answer: 119</strong></p><div class="trap-box"><strong>Trap:</strong> Surjections require inclusion-exclusion, not 5^7.<div class="key-concept"><strong>Key Concept:</strong> Injections = permutations; surjections = inclusion-exclusion
Correct Answer: 119