Question

Kanchan has $10$ friends of 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 is which she can invite five friends is

• A. ${}^8{C}_5$
• B. $2{\times }^8{C}_3$
• C. ${}^{10}{C}_5-2{\times }^8{C}_4$
• D. none of these

Answer 1

Answer: (B), (C)

$2{\times }^8{C}_3$
${}^{10}{C}_5-2{\times }^8{C}_4$

The number of ways of inviting, with the couple not included is ${}^8{C}_5$. The number of ways of inviting with the couple included, is ${}^8{C}_3$. Therefore the required number of ways is
${}^8{C}_5{+}^8{C}_3{=}^8{C}_3{+}^8{C}_3\left({\because }^8{C}_5{=}^8{C}_3\right)$
Also,
${}^{10}{C}_5-2{\times }^8{C}_4=\frac{10!}{5!5!}-2\times \frac{8!}{4!4!}$
$=\frac{10\times 9\times 8\times 7\times 6}{120}-2\times \frac{8\times 7\times 6\times 5}{24}$
$=9\times 4\times 7-140$
$=112=2\times \frac{8!}{3!5!}$