Question

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

• A. $112$
• B. $140$
• C. $164$
• D. $146$

Answer 1

The correct option is B

$140$

Explanation for the correct option:

Finding the number of ways of selecting:

Given, Nandan gives a dinner party to six guests out of ten

Total number of ways of selection of $6$ friends out of $10$ $$\begin{gathered} ={}^{10}C_6 \\ =\frac{10\times 9\times 8\times 7\times 6\times 5}{6\times 5\times 4\times 3\times 2\times 1}\left[\because {}^{n}C_r=\frac{n\times \left(n-1\right)\times \left(n-2\right)\times \dots \dots \times \left(n-r+1\right)}{r!}\right] \\ =210 \end{gathered}$$

Let us now find the possibilities where the two friends are in together.

If two friends are in together then we have to select four other friends from remaining eight.

Total number of possibilities where $2$ friends are in together is

$$\begin{gathered} ={}^{8}C_4 \\ =\frac{8\times 7\times 6\times 5}{4\times 3\times 2\times 1} \\ =70 \end{gathered}$$

Therefore, if two friends will not attend the party together, then the number of ways$=210-70=140$

Hence, the correct answer is option (B).