Difference between revisions of "2010 AMC 10A Problems/Problem 23"

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== Problem ==
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#REDIRECT [[2010_AMC_12A_Problems/Problem_19]]
Each of <math>2010</math> boxes in a line contains a single red marble, and for <math>1 \le k \le 2010</math>, the box in the <math>k\text{th}</math> position also contains <math>k</math> white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let <math>P(n)</math> be the probability that Isabella stops after drawing exactly <math>n</math> marbles. What is the smallest value of <math>n</math> for which <math>P(n) < \frac{1}{2010}</math>?
 
 
 
<math>\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005</math>
 
 
 
== Solution ==
 
=== Solution 1 ===
 
The probability of drawing a white marble from box <math>k</math> is <math>\frac{k}{k+1}</math>. The probability of drawing a red marble from box <math>n</math> is <math>\frac{1}{n+1}</math>.
 
 
 
The probability of drawing a red marble at box <math>n</math> is therefore
 
 
 
<center>
 
<math>\frac{1}{n+1} \left( \prod_{k=1}^{n-1}\frac{k}{k+1} \right) < \frac{1}{2010}</math>
 
 
 
<math>\frac{1}{n+1} \left( \frac{1}{n} \right) < \frac{1}{2010}</math>
 
 
 
<math>(n+1)n > 2010</math>
 
</center>
 
 
 
It is then easy to see that the lowest integer value of <math>n</math> that satisfies the inequality is <math>\boxed{45\ \textbf{(A)}}</math>.
 
 
 
=== Solution 2 ===
 
 
 
Using the first few values of <math>n</math>, it is easy to derive a formula for <math>P(n)</math>. The chance that she stops on the second box (<math>n=2</math>) is the chance of drawing a white marble then a red marble: <math>\frac{1}2 * \frac{1}3</math>. The chance that she stops on the third box (<math>n=3</math>) is the chance of drawing two white marbles then a red marble:<math>\frac{1}2 * \frac{2}3 * \frac{1}4</math>. If <math>n=4</math>, <math>P(n) = \frac{1}2 * \frac{2}3 * \frac{3}4 * \frac{1}5</math>.
 
 
 
Cross-cancelling in the fractions gives <math>P(2) = </math>\frac{1}{2*3}<math>, </math>P(3) = <math>\frac{1}{3*4}%, and </math>P(4) = <math>\frac{1}{4*5}</math>. From this, it is clear that <math>P(n) = \frac{1}{(n)(n+1)}</math>. (Alternatively, <math>P(n) = \frac{(n-1)!}{(n+1)!}</math>.)
 
 
 
<math>\frac{1}{(n+1)(n)} < \frac{1}{2010}</math>
 
 
 
The lowest integer that satisfies the above inequality is 45.
 
 
 
== See also ==
 
{{AMC10 box|year=2010|num-b=22|num-a=24|ab=A}}
 
 
 
[[Category:Introductory Combinatorics Problems]]
 

Latest revision as of 12:28, 26 May 2020

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