text: 1. ## Probability contest problem
  
  Professsor Gamble buys a lottery ticket, which requires that he pick six different integers from, 1 through 46, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property- the sum of the base ten logarithms is an integer. What is the probability that Professor Gamble holds the winner ticket?
  
  The answer is 1/4 and I got it after an hour of guessing and checking.
  I summarized how I reached the answer, but I wanted to check if it is reasonable. I'm starting to see my solutions are too sloppy, and they could be wrong even though I got the correct answer. Could someone show me a way to solve this with less guessing?
  
  Vicky.
  
  2. Hello, Vicky!
  
  Professsor Gamble buys a lottery ticket, which requires that he pick six different integers from 1 -46, inclusive.
  He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer.
  It so happens that the integers on the winning ticket have the same property:
  the sum of the base ten logarithms is an integer.
  What is the probability that Professor Gamble holds the winner ticket?
  
  Suppose the winning numbers are: .$\displaystyle a,b,c,d,e,f$
  
  The sum of the six logs is an integer: .$\displaystyle \log a + \log b + \log c + \log d + \log e + \log f \:=\:k$
  
  This means that the log of the product of the numbers is an integer: .$\displaystyle \log(abcde\!f) \:=\:k$
  
  . . Hence: .$\displaystyle abcde\!f \:=\:10^k$
  
  That is, the product of the 6 different numbers is a power of 10.
  
  I found only four cases:
  
  . . $\displaystyle \begin{array}{ccc}1\cdot2\cdot4\cdot5\cdot10\cdot2 5 &=& 10^4 \\ 1\cdot2\cdot5\cdot10\cdot25\cdot40 &=&10^5 \\ 1\cdot4\cdot5\cdot10\cdot20\cdot25 &=& 10^5 \\ 1\cdot5\cdot10\cdot20\cdot25\cdot40 &=& 10^6 \end{array}$
  
  Therefore, the Professor's probability of winning is $\displaystyle \tfrac{1}{4}$
  
  ### "professor gamble" solution
  
  Click on a term to search for related topics.
explanation: The text extract contains a math problem related to probability and logarithms. It describes a scenario where Professor Gamble buys a lottery ticket and picks six numbers such that the sum of their base-ten logarithms is an integer. The problem asks for the probability that Professor Gamble holds the winning ticket, given that the winning numbers also have the same property. The extract also includes a detailed solution to the problem, showing that there are four possible sets of numbers that satisfy the condition, leading to a probability of 1/4.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 1. ## Probability contest problem
  
  Professsor Gamble buys a lottery ticket, which requires that he pick six different integers from, 1 through 46, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property- the sum of the base ten logarithms is an integer. What is the probability that Professor Gamble holds the winner ticket?
  
  The answer is 1/4 and I got it after an hour of guessing and checking.
  I summarized how I reached the answer, but I wanted to check if it is reasonable. I'm starting to see my solutions are too sloppy, and they could be wrong even though I got the correct answer. Could someone show me a way to solve this with less guessing?
  
  Vicky.
  
  2. Hello, Vicky!
  
  Professsor Gamble buys a lottery ticket, which requires that he pick six different integers from 1 -46, inclusive.
  He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer.
  It so happens that the integers on the winning ticket have the same property:
  the sum of the base ten logarithms is an integer.
  What is the probability that Professor Gamble holds the winner ticket?
  
  Suppose the winning numbers are: .$\displaystyle a,b,c,d,e,f$
  
  The sum of the six logs is an integer: .$\displaystyle \log a + \log b + \log c + \log d + \log e + \log f \:=\:k$
  
  This means that the log of the product of the numbers is an integer: .$\displaystyle \log(abcde\!f) \:=\:k$
  
  . . Hence: .$\displaystyle abcde\!f \:=\:10^k$
  
  That is, the product of the 6 different numbers is a power of 10.
  
  I found only four cases:
  
  . . $\displaystyle \begin{array}{ccc}1\cdot2\cdot4\cdot5\cdot10\cdot2 5 &=& 10^4 \\ 1\cdot2\cdot5\cdot10\cdot25\cdot40 &=&10^5 \\ 1\cdot4\cdot5\cdot10\cdot20\cdot25 &=& 10^5 \\ 1\cdot5\cdot10\cdot20\cdot25\cdot40 &=& 10^6 \end{array}$
  
  Therefore, the Professor's probability of winning is $\displaystyle \tfrac{1}{4}$
  
  ### "professor gamble" solution
  
  Click on a term to search for related topics.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math problem related to probability and logarithms. It describes a scenario where Professor Gamble buys a lottery ticket and picks six numbers such that the sum of their base-ten logarithms is an integer. The problem asks for the probability that Professor Gamble holds the winning ticket, given that the winning numbers also have the same property. The extract also includes a detailed solution to the problem, showing that there are four possible sets of numbers that satisfy the condition, leading to a probability of 1/4.
  
  Label: question-and-answer