If you have decided to get a 20chocolate box and there are 1

If you have decided to get a 20-chocolate box and there are 19 different possible chocolates you can choose. How many ways are there to fill the box.

Solution

In case we are considering the arrangement of chocolates within the box also as a \'way\' then since each hole in the box can be filled by any of the 19 chocolates thus, total ways in which box can be filled is:

19x19x19x............... (20 times) = 19²⁰

In case we are not bothered with the permutation of the chocolates within the box then we can use the following formla to calculate different ways of filling the box with different kind of chocolates:

^(n+v-1)C_v-1 where n is the number of ways we want to arrnge v items i.e. we re considering the arrangement A A B and A B A as same where n=3 and v=2. Thus, 20 chocolates can be put in following ways:

^20+19-1C_19-1 =   ³⁸C₁₈