There are $3$ types of sandwiches, namely chicken (C), fish (F) and ham (H), available in a restaurant. A boy wishes to place an order of $6$ sandwiches. Assuming that there is no limit in the supply of sandwiches of each type, how many such orders can the boy place?
My Attempt:
I know that this is a stars and bars problem and the solution is $8 \choose 2$. But why can't it simply be $3^6$ where each of the $6$ sandwich places has $3$ choices to fill as repetition is allowed. I know I am missing something trivial.
Let us take Chicken and Ham sandwich and total order of 3. You are wondering why it should not be $2^3$ and why it is ${(3+2-1)\choose (2-1)}$
The $2^3=8$ choices that you mention are
AAA - 3 Chicken sandwich
AAB - 2 Chicken and 1 Ham
ABA
BAA
ABB - 2 Ham and 1 Chicken
BAB
BBA
BBB - 3 Ham
The total is $2^3$ but the chocies are merely ${(3+2-1)\choose (2-1)}$ = ${4\choose1}=4$