Relations & Functions
Number of Functions
Grade 12
Question:
<p>The number of functions \(f : \{a_1, a_2, \ldots, a_{10}\} \to \{b_1, b_2, \ldots, b_5\}\) is</p>
<p>(a) \(10^5\)</p>
<p>(b) \(5^{10}\)</p>
<p>(c) \(\frac{10!}{5!}\)</p>
<p>(d) \(5!\)</p>
Step-by-Step Solution
Key Concept: The number of functions from a finite set with <i>m</i> elements to a finite set with <i>n</i> elements is $n^m$, since each element in the domain can independently map to any of the <i>n</i> elements in the codomain.
<p><strong>Step 1:</strong> The domain has 10 elements: $\{a_1, a_2, \ldots, a_{10}\}$.</p><p><strong>Step 2:</strong> The codomain has 5 elements: $\{b_1, b_2, \ldots, b_5\}$.</p><p><strong>Step 3:</strong> For each of the 10 elements in the domain, we have 5 choices in the codomain.</p><p><strong>Step 4:</strong> By the multiplication principle, the total number of functions is $5 \times 5 \times 5 \times \cdots \times 5$ (10 times) = $5^{10}$.</p><p>∴ Answer is (b) $5^{10}$.</p>
Correct Answer: B