Counting - Case Work

SOURCE:COMPETITION
Number of Problems: 2. : (Book)

 

 

1.
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According to the standard convention for exponentiation, 2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.

If the order in which the exponentiations are performed is changed, how many other values are possible?

mathrm{(A) } 0qquad mathrm{(B) } 1qquad mathrm{(C) } 2qquad mathrm{(D) } 3qquad mathrm{(E) } 4


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2.
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How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

mathrm{(A) } 41qquad mathrm{(B) } 42qquad mathrm{(C) } 43qquad mathrm{(D) } 44qquad mathrm{(E) } 45

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