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# How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit?

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How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit?

Sep 15, 2018

#1
+2

How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit?

Ill add the word positive.  Smallest positive integers.

343 base10

7^3=343

so    343 base10     =     1000 base 7

There are a total of 343 numbers.

Lets think about the base 7 number. And first I am only going take a look at the ones less than 1000 base 7

The 7^2 digit can be 0 to 6 that is 7 digits

The 7^1 digit can be 0 to 6

The units digit can be 0 to 6 so long as the number is less than 1000 base7

How many have numbers less than 1000 base 7 have NO 4 or 5.

Each digit can be 0,1,2,3 or 6

5*5*5 = 125

[I have actually included 000 and not included 1000 but one in one out makes no difference]

So that is 125  that do not have a 4 or a 5

343- 125 = 218

So I think 218 do have a either a 4 or a 5 or both.

Sep 15, 2018
edited by Guest  Sep 15, 2018
edited by Melody  Sep 15, 2018
edited by Melody  Sep 15, 2018
edited by Melody  Sep 15, 2018
edited by Melody  Sep 15, 2018
#2
+2

I don't think you got this one quite right Melody.

Consider the 1 digit numbers.  The digit can't be 0, 4, 5.  That leaves 4 possible

2 digit numbers have 4C1 x (5C1) = 20 possibilities

3 digit numbers have 4C1 x (5C1)2 = 100 possibilities

There is 1 4 digit number, 1000, and it is counted

This gives 4 + 20 + 100 + 1 = 125 numbers

Oh, duh.  Ok, These are the numbers that don't match what you want.

We have to subtract this from the 343 and we get

343 - 125 = 218

Just like Melody got.  Sorry for the confusion.

Sep 15, 2018
edited by Rom  Sep 15, 2018
#4
+2

Hi Rom, I think you have interpreted the question incorrectly.

The question says we must use a 4 or a 5 or both.

If it is only one digit there are only 2 possibilities.

It must be 4 or 5.  So I think that you have answered a different question.

I tried doing it a different way but I still got my original answer. :/

Melody  Sep 15, 2018
edited by Melody  Sep 15, 2018
edited by Melody  Sep 15, 2018
#6
+1

That is ok Rom, I am glad we ended up on the same page :)

Melody  Sep 15, 2018
#5
+1

L:ets try again.

1000 does not have a 4 or a 5 so it is no good.

I want to know how many 3 digit base 7 numbers have at least one 4 or at least one 5

4@@         6       *3     =18

4@#        6*5   *3        =90

44@        6      *3       = 18

444                                 1

total  =  127

same for 5s   = 127

then

454            3

455             3

45*             5*3!=30

total = 36

one or the other or both = 2*127-36 = 218     So 218 have either a 4 or a 5 or both.

There are 343 numbers altogether so that just leaves 125 who do not have either one.

Sep 15, 2018