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 $= \,{}^8C_1 = 8$
(ii) $A, B, D$ selected and $C$ not selected
$\therefore $ Required ways $= \,{}^8C_1 = 8$
(iii) $A, B$ selected and $C, D$ not selected
$\therefore $ Required ways $= \,{}^8C_2 = 28$
(iv) $D$ selected and $A, B, C$ not selected
$\therefore $ Required ways $= \,{}^8C_3 = 56$
(v) $C$ selected and $A, B, D$ not selected
$\therefore $ Required ways $= \,{}^8C_3 = 56$
(vi) None of $A, B, C, D$ selected
$\therefore $ Required ways $=\,{}^8C_4 = 70$
Hence, the total number of ways for the problem
$= 8 + 8 + 28 + 2(56) + 70 = 226$