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 partly of 3 ladies and 3 gentlemen so that there are 3 of man's relative and 3 of the wife's relatives?
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This is a question on combination.
There are 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 ways
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