Question #14178

How many 7 digit numbers greater than 5,000,000 can be generated using the digits {3,4,5,5,6,6,6} if each of the seven digits must be used?
I can not understand his problem.Please someone help me UNDERSTAND HOW TO WORK THE PROBLEM and not just give an answer. ( i already know the answer just dont know how to get it)

Expert's answer

We are supposed to create numbers using digits. For example we can create numbers:

5566634 or change some 5566643 or 556634.. the question is how many different combinations of these digits we can generate (number should be more 500,000,000) thus we use formulas from Statistic and Probability Theory

There are a total of 7! / (3!*2!) ways of arranging those 7 numbers.

From this we must subtract the cases where the 3 or the 4 are first.

Cases with 3 first = 6! / (2!*3!)

Cases with 4 first = 6! / (2!*3!)

Answer = 7! / (3!*2!) - 2* 6! / (2!*3!)

= 420 - 2*60

= 300

5566634 or change some 5566643 or 556634.. the question is how many different combinations of these digits we can generate (number should be more 500,000,000) thus we use formulas from Statistic and Probability Theory

There are a total of 7! / (3!*2!) ways of arranging those 7 numbers.

From this we must subtract the cases where the 3 or the 4 are first.

Cases with 3 first = 6! / (2!*3!)

Cases with 4 first = 6! / (2!*3!)

Answer = 7! / (3!*2!) - 2* 6! / (2!*3!)

= 420 - 2*60

= 300

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