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 26x26x26x10x10x10 = 26 3 x 10 3 = 17,576,000. From this one style of plate, there are many sequences.

How many sequences of 4 things can be formed from 8 different things with replacement and order is important?

Explanation / Answer

from above as for first thing we have 8 choices again as with replacement for second, third and fourth thing also we have 8 choices.

hence total sequencces =8*8*8*8=4096

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