1.2 Selections

Suppose that a child is offered a bag containing three types of sweets, aniseed drops $(A)$, butter mints $(B)$ and cherry drops $(C)$. In how many ways can the child select two sweets?

This problem is not defined well, since we are not told:

• whether or not the same type of sweet can be selected twice, i.e. is $AA$ allowed?
• whether the order matter, i.e. is $AB$ to be treated the same as $BA$, or not?

There are 4 cases to consider:

1. Repeats are allowed and order matters. This gives 9 possibilities:
   • $AA,AB,AC,BA,BB,BC,CA,CB,CC$
   • Permutations with Unlimited Repetition
2. Repeats are allowed and order doesn’t matter. This gives 6 possibilities:
   • $AA,AB,AC,BB,BC,CC$
   • Combinations with Unlimited Repetition
3. Repeats are not allowed and order matters. This gives 6 possibilities
   • $AB,AC,BA,BC,CA,CB$
   • Permutations
4. Repeats are not allowed and order doesn’t matter. This give 3 possibilities
   • $AB,AC,BC$
   • Combinations