Relatives of a man comprises 4 ladies and 3 gentlemen and 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 manquotesingle s relative and 3 of the wifequotesingle s relatives.

Question

Relatives of a man comprises 4 ladies and 3 gentlemen and 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 relative and 3 of the wife's relatives.

DivineJEE · Accepted Answer

3 gentlemen and 3 ladies may be chosen as follows \ll Man Wife \ Case 1: 3(L) 3(G) \ Case 2: 2(L), 1(G) 2(G), 1(L) \ Case 3: 1(L), 2(G) 2(L), 1(G) \ Case 4: 3(G) 3(L) Required number of ways of inviting 3 ladies and 3 gentlemen to the party = ^4 C_3 x ^4 C_3+( ^4 C_2 x ^3 C_1)( ^3 C_1 x ^4 C_2) \ +( ^4 C_1 x ^3 C_2)( ^3 C_2 x ^4 C_1)+ ^3 C_3 x ^3 C_3 \ =16+324+144+1=485 .

Explanation

For (2 leq r leq n,binom{n}{r} + 2binom{n}{r - 1} + binom{n}{r - 2} =)

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