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 Y 2 2 sequences For each of the Y 2 2 sequences formed from 2 draws, there are Y ways the 3rd draw can occur forming Y 3 3 sequences Generalizing, there are Y X sequences formed by drawing X times from Y different things with replacement Example The number of state license plates can be made with 3 letters followed by 3 numbers is 26 x 26 x 26 x 10 x 10 x 10 26 3 3 x 10 3 3 17,576,000 From this one style of plate, there are many sequences How many sequences of 3 things can be formed from 7 different things with replacement and order is important

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,

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 Y $^{2}$ sequences. For each of the Y $^{2}$ sequences formed from 2 draws, there are Y ways the 3rd draw can occur forming Y $^{3}$ sequences. Generalizing, there are Y $^{X}$ sequences formed by drawing X times from Y different things with replacement.

Example: The number of state license plates can be made with 3 letters followed by 3 numbers is 26 x 26 x 26 x 10 x 10 x 10 = 26 $^{3}$ x 10 $^{3}$ = 17,576,000. From this one style of plate, there are many sequences.

How many sequences of 3 things can be formed from 7 different things with replacement and order is important?

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