Question

Ashok has $14$ friends including two married couples. He wants to invite $6$ persons for his wedding. Then the total number of ways to invite $6$ persons for the wedding such that

• A. only one couple can be invited is $2\times {}^{10}{C}_4$
• B. married couples refuse to come separately is $3\times {}^{10}{C}_4+{}^{10}{C}_8$
• C. both couples should not be invited together is $3\times {}^{10}{C}_4$
• D. married persons should come separately is ${}^{14}{C}_6-(2\times {}^{10}{C}_4+{}^{10}{C}_2)$

Answer 1

Answer: (B), (C), (D)

married persons should come separately is ${}^{14}{C}_6-(2\times {}^{10}{C}_4+{}^{10}{C}_2)$

Total number of ways to invite $6$ persons without any restrictions $={}^{14}{C}_6$
Number of ways to invite $6$ persons such that none of the couple is invited $={}^{10}{C}_6$
Number of ways to invite $6$ persons such that any one couple is invited $=2\times {}^{10}{C}_4$
Number of ways to invite $6$ persons such that both couples are invited $={}^4{C}_4\times {}^{10}{C}_2={}^{10}{C}_2$

$\left(1\right)$ Number of ways to invite $6$ persons such that only one couple should be invited $=$ no couple invited $+$ any one couple invited
$={}^{10}{C}_6+2\times {}^{10}{C}_4=3\times {}^{10}{C}_4(\because {}^n{C}_r={}^n{C}_{n-r})$

$\left(2\right)$ Number of ways to invite $6$ persons if couple refused to come separately $=$ no couple invited $+$ any one couple invited $+$ both couples invited
$={}^{10}{C}_6+2\times {}^{10}{C}_4+{}^{10}{C}_2=3\times {}^{10}{C}_4+{}^{10}{C}_8$

$\left(3\right)$ Number of ways to invite $6$ persons such that both couples should not be invited $=$ no couple invited $+$ any one couple invited
$={}^{10}{C}_6+2\times {}^{10}{C}_4=3\times {}^{10}{C}_4$

$\left(4\right)$ Number of ways to invite $6$ persons such that married persons should come separately $=$ Total ways to invite $6$ persons $-$ number of ways of inviting couples
$={}^{14}{C}_6-(2\times {}^{10}{C}_4+{}^{10}{C}_2)$