Pulling six christmas crackers from a pool of six people **A family of six has six Christmas crackers to pull. Each person will pull two crackers, each with a different person. In how many different ways can this be done?** So far I have found $20$ possibilities, because the six members can be separated into two groups of $3$. Each group of 3 has one distinct way of pulling crackers, and there are ${6 \choose 3}$ ways of selecting two groups of 3, giving $20$ ways of cracking crackers. However I'm stuck on how to find more from here onwards. The desired answer is $70$.

You've got the right idea. However, the number of ways dividing $6$ people into two groups of three is not 20. It is 10. You may want to read more about it here.

The other case you're missing is when the six people form a cycle of their own - i.e. if you place the family members on a round table, each person forms a pair with their neighbour. The number of ways in which this is done is $\frac{5!}{2} = 60.$

Thus, the answer is $$10 + 60 = 70$$