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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
8 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
3 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
3 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
one year 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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