Question

A delegation of four friends to be selected from a group of $12$ friends. The number of ways, the delegation be selected if two particular friends refused to be together and two other particular friends wish to be together only in the delegation, is

• A. $226$
• B. $114$
• C. $156$
• D. $170$

Answer 1

Answer: (A)

Let us call the friends as $A,B,C,D$. Assume
$A,B$ wish to be together and $C,D$ don’t wish to be
together. Now, the following six cases arise.
(i) $A,B,C$ selected and $D$ not selected
$\therefore$ Number of ways $={}^8{C}_1=8$
(ii) $A,B,D$ selected and $C$ not selected
$\therefore$ Required ways $={}^8{C}_1=8$
(iii) $A,B$ selected and $C,D$ not selected
$\therefore$ Required ways $={}^8{C}_2=28$
(iv) $D$ selected and $A,B,C$ not selected
$\therefore$ Required ways $={}^8{C}_3=56$
(v) $C$ selected and $A,B,D$ not selected
$\therefore$ Required ways $={}^8{C}_3=56$
(vi) None of $A,B,C,D$ selected
$\therefore$ Required ways $={}^8{C}_4=70$
Hence, the total number of ways for the problem
$=8+8+28+2(56)+70=226$