Question

A woman has $11$ close friends. The number of ways in which she can invite $5$ of them to dinner, if two particular of them are not on speaking terms and will not attend together, is

• A. ${}^{11}{C}_5-{}^9{C}_3$
• B. ${}^9{C}_5+2\times {}^9{C}_4$
• C. $3\times {}^9{C}_4$
• D. ${}^9{C}_4$

Answer 1

Answer: (A), (B), (C)

$3\times {}^9{C}_4$

Total number of possible selections
$={}^{11}{C}_5$
Number of ways in which two particular women attend together $={}^2{C}_2\times {}^9{C}_3={}^9{C}_3$
Required number of ways
$={}^{11}{C}_5-{}^9{C}_3={}^9{C}_4[\frac{11\times 10}{6\times 5}-\frac{4}{6}]=3\times {}^9{C}_4$


Alternate solution:
Let two particular women be $W_1$ and $W_2$
If $W_1$ does not attend the party while $W_2$ does, number of ways $={}^1{C}_1\times {}^9{C}_4={}^9{C}_4$
If $W_2$ does not attend the party while $W_1$ does, number of ways $={}^1{C}_1\times {}^9{C}_4={}^9{C}_4$
If neither of $W_1$ and $W_2$ attends the party, number of ways $={}^9{C}_5$
Required total number of ways
$={}^9{C}_4+{}^9{C}_4+{}^9{C}_5=3\times {}^9{C}_4$