After reviewing the video on Variations with repetition, we were instructed to think of it as “the number of variations with repetitions when picking p-many elements out of n elements is equal to n to the power of p“. The Quiz question asked about a combination lock code consisting of a 4 letter word (i.e. total number of elements (n)) using letters A, B, and C (i.e. p-many elements). Why is the answer to the quiz is 3^4? Shouldn’t the correct answer be 4^3?
n = 4 (total number of elements)
p = 3 (p-many elements)
The simplest way to think about this is the following. The passcode consists of 4 letters, so there are 4 “positions” in the word -> _ _ _ _.
We can use any of the letters A, B and C as a first letter of the passcode, so there are 3 ways to fill out position 1 -> 3 _ _ _. Then, regardless what letter we choose, we still have 3 options for the second letter, 3 3 _ _ and so on. Hence, we have 3 options for reach additional position, regardless of what we’ve chosen so far.
Hence, 3*3*3*3 = 3^4.