What is the value of \({^{81}P}_2-{^{81}C}_2\)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3240
\({^nP}_r\) counts arrangements (order matters) and \({^nC}_r\) counts selections (order does not matter).
\({^{81}P}_2=81\times80=6480\)
\({^{81}C}_2=\displaystyle\frac{81\times80}{2}=3240\)
\({^{81}P}_2-{^{81}C}_2=6480-3240=3240\)
Shortcut:
Each selection of 2 can be arranged in \(2!=2\) ways, so \({^{n}P}_2=2\times{^{n}C}_2\) and the difference is \({^{81}C}_2=3240\) itself.