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7 relatives of a man comprises 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 man's relatives and 3 of the wife's relatives?

7 relatives of a man comprises 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 man's relatives and 3 of the wife's relatives?

Grade:upto college level

4 Answers

Navjyot Kalra
askIITians Faculty 654 Points
9 years ago
Hello Student,
Please find the answer to your question
There four possibilities:
(i) 3 ladies from husband’s side and 3 gentlemen from wife’s side.
No. of ways in this case
= 4 C3 x 4 C3 = 4 x 4 = 16
(ii) 3 gentlemen from husband’s side and 3 ladies from wife’s side.
No. of ways in this case = 3 C3 x 3 C 3 = 1 x 1 = 1
(iii) 2 ladies and one gentleman from husband’s side and lady and 2 gentlemen from wife’s side.
No. of ways in this case
= (4 C4 x 3 C1) x (3 C1 x 4 C2) = 6 x 3 x 3 x 6 = 324
(iv) One lady and 2 gentlemen from husband’s side and 2 ladies and one gentlemen from wife’s side.
No. of ways in this case
= (4 C1 x 3 C2) x (3 C2 x 4 C1) = 4 x 3 x 3 x 4 = 1444
Hence the total no. of ways are
= 16 + 1 + 324 + 144 = 485.

Thanks
navjot kalra
askIITians Faculty
SUKRITi c
14 Points
5 years ago
I think it should be 215 as it is 16 +54 +144+1 = 215 and not 16+324+144+1 =  485
Your 324  which is ( 4C4 x 3C1 ) x ( 4C2 x 3C1 )  should it not be (4C2 x 3C1) x (4C2 x 3C1 ) which is 54 ?
 
Please do correct me if I’m wrong as I am still an aspirant to make it to IIT  , God Willing.
SUKRITi c
14 Points
5 years ago
I was goingg wrong in calculation. Now I got the corrcet answer 485.
 
But your (4C4 x 3C1) x (4C2 x 3C1)  should be (4C2 x 3C1 ) x (4C2 x 3C1 ) . May be a typographical error in your solution  where intended 4C2 was erroneously typed as 4C4 .. With 4C4  you will get 54 and not 324
Rishi Sharma
askIITians Faculty 646 Points
3 years ago
Dear Student,
Please find below the solution to your problem.

There four possibilities:
(i) 3 ladies from husband’s side and 3 gentlemen from wife’s side.
No. of ways in this case
= 4 C3 x 4 C3 = 4 x 4 = 16
(ii) 3 gentlemen from husband’s side and 3 ladies from wife’s side.
No. of ways in this case = 3 C3 x 3 C 3 = 1 x 1 = 1
(iii) 2 ladies and one gentleman from husband’s side and lady and 2 gentlemen from wife’s side.
No. of ways in this case
= (4 C4 x 3 C1) x (3 C1 x 4 C2) = 6 x 3 x 3 x 6 = 324
(iv) One lady and 2 gentlemen from husband’s side and 2 ladies and one gentlemen from wife’s side.
No. of ways in this case
= (4 C1 x 3 C2) x (3 C2 x 4 C1) = 4 x 3 x 3 x 4 = 1444
Hence the total no. of ways are
= 16 + 1 + 324 + 144 = 485.

Thanks and Regards

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