Combinatorics dance class

Question

I have a problem that ím trying to solve and I am not completely sure if my answer is correct, I tried looking for it on the web but i cant find a problem quite like it.

I am translating the question into English, and im doing my best to translate it correct, but im sorry if there are some stupid grammar mistakes.

„During a dance class 4 pairs (a pair consists of one man and one woman) are chosen from 4 men and 7 women. Romeo and Juliet are students in this class, what is the probability that the two will form a pair.

My answer would be

$$ \frac{\binom{6}{3}3!}{ \binom{7}{4}4!} $$

My thought process is as follows,

There is only one possibility two choose Rome and Juliet, and 1 Possibility to choose the 3 men, so im not writing them down. But there are 6C3 possible ways to choose the 3 women left, and 3! possibilities to assign them to the 3 men.

And there are 7C4 ways to choose the 4 women in total and then 4! Ways to form different pairs.

Is this correct? And if not, where did i make a mistake.

Many thanks for reading this and helping me out

Ps

I tried writing it down in mathjax but wasn’t very successful with that - sorry

Answer 1

Yes, your reasoning is correct.

Replaying your argument . . .

There are ${\large{\binom{7}{4}}}4!$ possible results for the selection of $4$ dance pairs.

Explanation:

• There are ${\large{\binom{7}{4}}}$ possible choices for the $4$ women.$\\[4pt]$
• Once the $4$ women are chosen, there are $4!$ ways to match them with the $4$ men.

There are ${\large{\binom{6}{3}}}3!$ possible results for the selection of $4$ dance pairs such that Romeo is paired with Juliet.

Explanation:

• There are ${\large{\binom{6}{3}}}$ possible choices for the $3$ women other than Juliet.$\\[4pt]$
• Once those $3$ women are chosen, there are $3!$ ways to match them with the $3$ men other than Romeo.

Therefore the probability that Romeo is paired with Juliet is $$\frac{\binom{6}{3}3!}{\binom{7}{4}4!}=\frac{1}{7}$$

So that was your argument, and it's fine, but the answer can be found in a much simpler way . . .

After all, Romeo will be paired with some woman, and there is no bias, hence, since there are $7$ women, the probability that Romeo is paired with Juliet is ${\large{\frac{1}{7}}}$.