Sam and Jessica are invited to a dance. If there are 7 men and 7 women

Welcome to the Gmat Club.

The easiest way to solve this problem is to calculate opposite probability and subtract this value from 1:
The opposite probability is that Sam and Jessica WILL be chosen to lead \(=\frac{1}{7}*\frac{1}{7}=\frac{1}{49}\).
So, the probability that they will NOT be chosen to lead \(= 1-\frac{1}{49}=\frac{48}{49}\).

Answer: E.

The way you are doing is wrong because \(\frac{36}{49}\) is the probability that in the leading pair won't be neither Sam nor Jessica. But in the leading pair can be Sam with any woman but Jessica and can be Jessica with any man but Sam.

So, if you are doing this way you should count \(\frac{7}{7}*\frac{6}{7}\) probability of pairing any man (\(\frac{7}{7}\)) with any woman but Jessica (\(\frac{6}{7}\)) PLUS \(\frac{1}{7}*\frac{6}{7}\) probability of pairing Jessica (\(\frac{1}{7}\)) with any man but Sam (\(\frac{6}{7}\)) \(=\frac{7}{7}*\frac{6}{7}+\frac{1}{7}*\frac{6}{7}=\frac{48}{49}\).

OR you can think about it this way: there are total \(7*7=49\) pairs possible, and only one pair is when Jessica and Sam are together, hence probability \(=\frac{49-1}{49}=\frac{48}{49}\)

OR: pairs without Jessica \(6*7=42\) and pairs with Jessica but without Sam \(1*6=6\), hence probability \(=\frac{42+6}{49}=\frac{48}{49}\).

Hope it helps.