\(\left({ }^{75} P_2-{ }^{75} C_2\right)=?\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2775
\({}^{n}P_2\) counts ordered pairs and \({}^{n}C_2\) unordered ones; each unordered pair can be arranged in 2 ways, so \({}^{n}P_2=2\times{}^{n}C_2\).
Ordered pairs: \({}^{75}P_2=75\times74=5550\)
Unordered pairs, half as many: \({}^{75}C_2=\displaystyle\frac{5550}{2}=2775\)
Difference: \({}^{75}P_2-{}^{75}C_2=5550-2775=2775\), the same as \({}^{75}C_2\)