Question 156377: How many 6 digit numbers can be made with any combination and permutation of 3 and 7? (e.g. 377777, 737777, 777333, 733733, 333337)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let's try to find the number of combinations that 3 and 7 can make up for a two digit number.
Let's try a smaller example. So let's use the numbers 3 and 7 to make a two digit number, three digit number, and a four digit number to see what is going on
So let's list ALL of the possible two digit numbers:
33
37
73
77
So for a two digit number, there are 4 possible combinations
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So let's list ALL of the possible three digit numbers:
333
337
373
377
733
737
773
777
So for a three digit number, there are 8 different combinations.
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So let's list ALL of the possible four digit numbers:
3333
3337
3373
3377
3733
3737
3773
3777
7333
7337
7373
7377
7733
7737
7773
7777
So for a four digit number, there are 16 different combinations.
Notice how the number of combinations keeps doubling (since 2 doubles to 4, 4 doubles to 8, 8 doubles to 16, etc...).
So if we write out a 5 digit number only using 3 and 7, then we will list out 32 different combinations (which is double of 16). And finally for a six digit number, there will be 64 different combinations (which is double of 32).
Note: the quicker way (ideally once you understand what's going on) is to use this formula  where  is the number of digits and  is the number of unique number combinations.
So in this case,  and  . So once again, we get 64 different combinations.
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Answer:
So there are 64 unique 6 digit numbers that can be generated from just the two values 3 and 7.
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