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

only one couple can be invited is

2 \times^{10} C_{4}

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married couples refuse to come separately is

3 \times^{10} C_{4} +^{10} C_{8}

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both couples should not be invited together is

3 \times^{10} C_{4}

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married persons should come separately is

^{14} C_{6} - (2 \times^{10} C_{4} +^{10} C_{2})

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Solution

The correct options are
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 $^{14} C_{6} - (2 \times^{10} C_{4} +^{10} C_{2})$
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 $=^{4} C_{4} \times^{10} C_{2} =^{10} C_{2}$

$(1)$ Number of ways to invite $6$ persons such that only one couple should be invited $=$ no couple invited $+$ any one couple invited
$=^{10} C_{6} + 2 \times^{10} C_{4} = 3 \times^{10} C_{4} (∵^{n} C_{r} =^{n} C_{n - r})$

$(2)$ Number of ways to invite $6$ persons if couple refused to come separately $=$ no couple invited $+$ any one couple invited $+$ both couples invited
$=^{10} C_{6} + 2 \times^{10} C_{4} +^{10} C_{2} = 3 \times^{10} C_{4} +^{10} C_{8}$

$(3)$ Number of ways to invite $6$ persons such that both couples should not be invited $=$ no couple invited $+$ any one couple invited
$=^{10} C_{6} + 2 \times^{10} C_{4} = 3 \times^{10} C_{4}$

$(4)$ 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
$=^{14} C_{6} - (2 \times^{10} C_{4} +^{10} C_{2})$

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