Math CentralQuandaries & Queries


Question from shabkhal, a parent:

Please help me with this prob:

If a four-digit number is chosen at random, what is the probability that the product of the digits is 12.


How many 4 digit numbers are there? The answer will depend on what you consider to be a 4 digit number. For example is 0025 a 4 digit number?

If the product of the digits of a 4 digit number is 12 then the digits must be 1, 1, 3 and 4 or 1, 1, 2 and 6.

How many 4 digit numbers can you form from 1, 1, 3 and 4? You can easily write them down and count them but let's do something more systematic. Suppose the question were "How many 4 digit numbers can you form from 4 distinct digits, say 1, 7, 3 and 4?" If I am going to form a 4 digit number from these digits then I have 4 choices for the first digit. Regardless of which digit I have chosen for the first digit I have 3 choices for the second digit. Again I have 2 choices for the third digit and 1 choice for the last digit. Hence there are 4 × 3 × 2 × 1 = 24 different 4 digit numbers I can form from 1, 7, 3 and 4.

If I know replace each 7 by a 1 then I lose half the numbers. For example 1374 and 7314 each become 1314. Hence there are 24/2 = 12 four digit numbers can you can form form 1, 1, 3 and 4.

Can you complete the problem now?


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