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Here, Friends of X = 4 Ladies + 3 Men

Friends of Y = 3 Ladies + 4 Men

Case-1: When 3 ladies from X and 3 men from Y will be invited.

So, no. of ways =

$

C(4,3) \times C(4,3) \\

= 4 \times 4 \\

= 16 \\

$

Case-2: When 2 ladies from X and 1 lady from Y will be invited with 1 man from X and 2 men from Y.

So, no. of ways =

$

C(4,2) \times C(3,1) \times C(3,1) \times C(4,2) \\

= 6 \times 3 \times 3 \times 6 \\

= 324 \\

$

Case-3: When 1 lady from X and 2 ladies from Y, and from 2 men from X and 1 man from Y will be invited.

So, no. of ways =

$

C(4,1) \times C(3,2) \times C(3,2) \times C(4,1) \\

= 4 \times 3 \times 3 \times 4 \\

= 144 \\

$

Case-4: When 0 lady from X means 3 men from X and 3 ladies from Y will be invited.

So, no. of ways =

$

C(3,3) \times C(3,3) \\

= 1 \times 1 \\

= 1 \\

$

$\therefore $ Total number of ways will be = 16 + 324 + 144 + 1

= 485