Suppose a sample space has things a, b, and c. Twice, draw from the sample space

Question

Suppose a sample space has things a, b, and c. Twice, draw from the sample space and replace. The possible sequences formed are {aa, ab, ac, ba, bb, bc, ca, cb, cc}. Now suppose there are Y different things. There are Y ways the first draw can occur. For each of the Y ways the first draw can occur, there are Y ways the second draw can occur, resulting in Y times Y, or Y2 sequences. For each of the Y2 sequences formed from 2 draws, there are Y ways the 3rd draw can occur forming Y3 sequences. Generalizing, there are YX sequences formed by drawing X times from Y different things with replacement. Example: The number of state license plates that can be made with 3 letters followed by 3 numbers is 26 x 26 x 26 x 10 x 10 x 10 = 263 x 103 = 17,576,000. From this one style of plate, there are many sequences. How many sequences of 5 things can be formed from 8 different things with replacement and order is important?

Explanation / Answer

In order to select every item, 8 things are available. Selecting 1 item from 8 things can be done in 8C1 ways = 8 ways.

total number of possible sequences are = 8^5 = 32768

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