Question 459567
a. Start with an X and letters can be repeated 
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Examples:  XAJQN, XLLTX, XXXXX 

Choose letter#1 1 way, choose letter#2 26 ways, 
choose letter#3 26 ways, choose letter#4 26 ways,
choose letter#5 26 ways.

That's 1×26×26×26×26 = 456976 ways

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b. No letter can be repeated
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Examples:  kYUDM, PIANO, VWXYZ

Choose letter#1 26 ways, choose letter#2 25 ways, 
choose letter#3 24 ways, choose letter#4 23 ways,
choose letter#5 22 ways.

That's 26×25×24×23×22 = 7893600 ways.

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c. Start and end with an X and letters can be repeated 
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Examples:  XEROX, XFFJX, XXXXX

Choose letter#1 1 way, choose letter#2 26 ways, 
choose letter#3 26 ways, choose letter#4 26 ways,
choose letter#5 1 way.

That's 1×26×26×26×1 = 17576 ways.

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d. Start with the letters BO(in that order) and letters can be repeated 
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Examples:  BOOZE, BOAST, BOBOO

Choose letter#1 1 way, choose letter#2 1 way, 
choose letter#3 26 ways, choose letter#4 26 ways,
choose letter#5 26 ways.

That's 1×1×26×26×26 = 17576 ways.

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e. Start and end with the letters BO(in that order) and letters can be repeated 
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Examples:  BOOBO, BOMBO, BOBBO

Choose letter#1 1 way, choose letter#2 1 way, 
choose letter#3 26 ways, choose letter#4 1 way,
choose letter#5 1 ways.

That's 1×1×26×1×1 = 26 ways.
 
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f. Start or end with the letters BO(in that order)if letters can be repeated
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We must use a formula here:

N(X or Y) = N(X) + N(Y) - N(X and Y), 
  where N( ) means "the number of".

Let X be "words beginning with BO", 
        calculated in part d, so N(X) = 17576
Let Y be "words ending with BO", 
        same as the results of part d, so N(Y) = 17576
Then "X and Y" will be "words beginning 
        and ending with BO", calculated 
        in part e, so N(X and Y) = 26

N(X or Y) = 17576 + 17576 - 26 = 35126

Edwin</pre>