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Question

7 relatives of a man comprise 4 ladies and 3 gentlemen.His wife has also 7 relatives, 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?


A

485

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B

500

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C

486

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D

102

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Solution

The correct option is A

485


Explanation for the correct option:

Step 1: List out all possible cases of invitation:

Both Men and Women have to invite their guests each, and there should be 3 ladies and 3 gentlemen at the party.

Then, the man can invite 3L,2L&1g,1L&2g,3g ,and women can invite (3g,2g&1L,1g&2L,3L) correspondingly.

Step 2: Solve each case to get the total number of ways:

Case I: A man sends invites to 3 of his ladies relatives, and the woman sends invites to 3 of her gentlemen relatives.

C34·C34=16

Case II: A man sends invites to 2 of his ladies relatives, 1 of his gentleman's relative and the woman sends invites to 2 of her gentlemen relatives and 1 of her lady relative.

C24·C13C13·C24=324

Case III: A man sends invites to 1 of his lady relative, 2 of his gentlemen relatives and the woman sends invites to1of her gentlemen relative and 2 of her ladies relatives.

C14·C23C23·C14=144

Case IV: A man sends invites to 3 of his gentlemen relatives and the woman sends invites to 3 of her ladies relatives.

C33·C33=1

Step 3: Sum up all the cases to get the required number of ways:

Total ways of invitation=16+324+144+1=485

Hence, option (A) is the correct answer.


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