MathDB

3

Part of 2002 AMC 10

Problems(3)

Exponentiation Order

Source:

2/11/2008
According to the standard convention for exponentiation, 2^{2^{2^2}} \equal{} 2^{\left(2^{\left(2^2\right)}\right)} \equal{} 2^{16} \equal{} 65,\!536. If the order in which the exponentiations are performed is changed, how many other values are possible?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 4 <span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4
Arithmetic Mean

Source:

2/26/2008
The arithmetic mean of the nine numbers in the set {9,99,999,9999,...,999999999} \{9,99,999,9999,...,999999999\} is a 9 9-digit number M M, all of whose digits are distinct. The number M M does not contain the digit <spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 8 <span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 8
Typo-ing a Six Digit Number

Source:

4/2/2013
Mary typed a six-digit number, but the two 11s she typed didn't show. What appeared was 20022002. How many different six-digit numbers could she have typed?
<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>8<spanclass=′latex−bold′>(C)</span>10<spanclass=′latex−bold′>(D)</span>15<spanclass=′latex−bold′>(E)</span>20<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>10\qquad<span class='latex-bold'>(D) </span>15\qquad<span class='latex-bold'>(E) </span>20
countingdistinguishability