Find Number of One-One Functions from \(A\) to \(B\)
📝 Question
Let:
\[ A=\{a,b,c\}, \quad B=\{-2,-1,0,1,2\} \]
Find the total number of one-one (injective) functions from \(A\) to \(B\).
✅ Solution
🔹 Step 1: Use Formula
If \(|A|=m\) and \(|B|=n\) with \(n \ge m\), then number of one-one functions is:
\[ {}^{n}P_{m}=\frac{n!}{(n-m)!} \]
(Permutation formula) :contentReference[oaicite:0]{index=0}
🔹 Step 2: Substitute Values
\[ |A|=3,\quad |B|=5 \]
\[ \text{Number of one-one functions} = {}^{5}P_{3} \] —
🔹 Step 3: Calculate
\[ {}^{5}P_{3}=\frac{5!}{(5-3)!}=\frac{5!}{2!} \]
\[ =\frac{5\times4\times3\times2\times1}{2\times1} \]
\[ =5\times4\times3=60 \] —
🎯 Final Answer
\[ \boxed{60} \]
🚀 Exam Shortcut
- Use formula: \(^{n}P_{m}\)
- Here: \(5P3 = 5 \times 4 \times 3\)
- Answer = 60
- Think: assign distinct images to each element