A) 112
B) 140
C) 164
D) none of these
Correct Answer: B
Solution :
Total number of friends \[=10\] Number of friends to be invited\[~=6\] Let A, B be the two friends who are not attend the party together. Case I: If only one of them attend the party. Case II: If none of A, B attend the party \[\therefore \]Total number of ways of inviting friends \[={{\,}^{10-2}}{{C}_{6}}\times {{\,}^{2}}{{C}_{0}}+{{\,}^{10-2}}{{C}_{5}}\times {{\,}^{2}}{{C}_{1}}\] \[={{\,}^{8}}{{C}_{6}}\times 1+{{\,}^{8}}{{C}_{5}}\times 2\] \[=28+112=140\] Note: (i) If there are n persons, r persons to be selected in which two particular persons never selected \[={{\,}^{n-r}}{{C}_{r}}\]ways. (ii) If two particular persons always be selected, then total number of selecting \[={{\,}^{n-2}}{{C}_{r-2}}\]ways.
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