Question

A lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends, if two of the friends will not attend the party together is?

• A. $112$
• B. $140$
• C. $164$
• D. None of these

Answer 1

The correct option is B

$140$

Finding the value if two of the friends will not the attend the party together:

It is given that the number of guests $=6$, total friends $=10$

The selection of $6$ guests from $10$ friends is:

$\begin{array}{rcl}{}^{10}{C}_6& =& \frac{10!}{6!\times 4!}\\ & =& 210\end{array}$

The selection of $2$ friends always will be:

$\begin{array}{rcl}{}^{2}C_{2}x{}^{8}C_4& =& 1x\frac{8!}{4!\times 4!}\\ & =& 70\end{array}$

$\therefore$Number of ways, to select $6$ out of $10$ friends when two will not attend the party together $=210–70$

$=\mathbf{}\mathbf{140}$

Hence, the correct answer is option (B).