Sets, Relations & Functions
General
Grade 11
Question:
<p>If <span class="math-inline">\(|X|=5\)</span>, <span class="math-inline">\(|Y|=7\)</span>, <span class="math-inline">\(\alpha=\)</span>#injections <span class="math-inline">\(X\to Y\)</span>, <span class="math-inline">\(\beta=\)</span>#surjections <span class="math-inline">\(Y\to X\)</span>. Find <span class="math-inline">\(\frac{\beta-\alpha}{5!}\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>α=⁷P₅=2520. β=5⁷-C(5,1)4⁷+C(5,2)3⁷-C(5,3)2⁷+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 5⁷.</div><div class="key-concept"><strong>Key Concept:</strong> Injections = permutations; surjections = inclusion-exclusion</div></div>
Correct Answer: 119