Kanchan has 10 friends among whom two are married to each other. She wishes to invite five of them for a party. If the married couples refuse to attend separately, then the number of different ways in which she can invite five friends is

The number of ways of inviting, with the couple not included, is ${ }^8{\mathrm{C}}_5$. The number of ways of inviting with the couple included, is ${ }^8{\mathrm{C}}_3$. Therefore the required number of ways is

${ }^8{\mathrm{C}}_5{+}^8{\mathrm{C}}_3{=}^8{\mathrm{C}}_3{+}^8{\mathrm{C}}_3\left({\because }^8{\mathrm{C}}_5{=}^8{\mathrm{C}}_3\right)$

Also,

$\begin{array}{c}{ }^{10}{\mathrm{C}}_5-2{\times }^8{\mathrm{C}}_4=\frac{10!}{5!5!}-2\times \frac{8!}{4!4!}\\ =\frac{10\times 9\times 8\times 7\times 6}{120}-2\times \frac{8\times 7\times 6\times 5}{24}\\ =9\times 4\times 7-140\\ =112=2\times \frac{8!}{3!5!}\end{array}$