Finding the combination of creating 2 pairs

Question

We have 5 men and 3 women. We are to create 2 pairs of man/female ( the order of pairs does not matter )

We can create one pair using basic formula for combinations using

$\binom{5}{1}\binom{3}{1}=15$

But how can we create two pairs?

i tried using

$\binom{5}{1}\binom{3}{1} + \binom{4}{1}\binom{2}{1}$ Since first pair has one man and one woman ,in the second pair we choose from one less man and woman. This equals 23 but that answer is incorrect.

// edit

i realised i cannot add it together and edited my formula into

$\binom{5}{1}\binom{3}{1}\binom{4}{1}\binom{2}{1}$ which is 120 but still incorrect

Answer 1

Multiply the following:

• Number of ways to choose $2$ out of $5$ men: $\binom52=10$
• Number of ways to choose $2$ out of $3$ women: $\binom32=3$
• Number of ways to split $2$ men and $2$ women into $2$ couples: $2$

The answer is $10\cdot3\cdot2=60$.

Answer 2

It is like arranging a mixed doubles tennis match.

The $4$ individuals can be selected in $\binom52\binom32 = 30$ ways

You can be paired with any one from the two of opposite gender,

thus $30\cdot\binom21 = 60$ ways