How many different 7 digit members are their sum of whose

I guess the question is:

How many different 7-digit numbers are there the sum of whose digits are even?

From the first glance I'd say exactly half of the number will have the sum of their digits odd and half even, why should there be more even than odd?

Consider first ten 7-digit integers: 1,000,000-1,000,009 - half has even sum and half odd;
Next ten 7-digit integers: 1,000,010-1,000,019 - also half has even sum and half odd;
And so on.

There are \(9*10^6\) 7-digit integers, half of it, \(45*10^5\) will have odd sum of their digits, or as crack700 wrote \(288*5^6\), (\(288*5^6=2^5*9*5^6=(9*5)*(2^5*5^5)=45*10^5\)).

Hope it's clear.
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I like Bunuel's approach, or you can think of things as follows. Let's start by ignoring the units digit:

* we have 9 choices for the first digit, and 10 choices for each remaining digit, so there are 9*10^5 choices for the first six digits.

* now the sum of the first six digits is either even or odd. If it's even, we need our last digit to be even, so we have 5 choices. If it's odd, we need our last digit to be odd, so we have 5 choices. So no matter how we choose our first six digits, we have 5 choices for our units digit, and the answer must then be 9 * 10^5 * 5 = 45* 10^5.
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My attempt:

7 digit number is of the form abcdefg. In this the sum of the digits would end up as a even number only if the following is true.

a) There number of even number among these numbers -- a, b, c, d, e, f and g should be either 1 or 3 or 5 or 7.

Also the number of even and odd single digit numbers are 5. (1,3,5,7,9 and 0, 2, 4, 6, 8)

To summarize, let us take the even digit to be e and odd digit to be o.

Hence the number abcdefg could be either

ooooooe OR ooooeee OR ooeeeee OR eeeeeee

Number of ways to choose a even or odd digit is 5.

Hence answer is 4*5^7.

Here is how i approached it.

Sum of the 7 digits is even if the number has 1 or 3 or 5 or 7 even digits
There are 5 even digits available 0,2,4,6,8 and 5 odd digits available 1,3,5,7,9. However the first digit in the 7 digit number should not be 0. Otherwise it will become a 6 digit number.

7E 0O = \(5^7-5^6\)
There are 5 even numbers available. So each of 7 the digits can be filled with an even number in \(5^7\) ways, but this also contains the numbers having 0 in the first digit location.
If first digit has a 0 , rest 6 digits can be filled with an even number in \(5^6\) ways

5E 2O = \(C^7_5*5^5*5^2-C^6_4*5^4*5^2\)

We can first pick the 5 spots having even numbers in \(C^7_5\) ways ,fill them each with even number in \(5^5\) ways and fill the remaining 2 digits with odd numbers in \(5^2\) ways. Again we need to subtract the numbers having 0 in the first digit location
If first digit has a 0 , we can pick the 4 spots out of the remaining 6 having even numbers in \(C^6_4\) ways ,fill them each with even number in \(5^4\) ways and fill the remaining 2 digits with odd numbers in \(5^2\) ways.

3E 4O = \(C^7_3*5^3*5^4-C^6_2*5^2*5^4\)

We can first pick the 3 spots having even numbers in \(C^7_3\) ways ,fill them each with even number in \(5^3\) ways and fill the remaining 4 digits with odd numbers in \(5^4\) ways. Again we need to subtract the numbers having 0 in the first digit location
If first digit has a 0 , we can pick the 2 spots out of the remaining 6 having even numbers in \(C^6_2\) ways ,fill them each with even number in 5^2 ways and fill the remaining 4 digits with odd numbers in \(5^4\) ways.

1E 6O = \(C^7_1*5^1*5^6-5^6\)

We can first pick the 1 spot having an even number in \(C^7_1\) ways ,fill it with an even number in \(5^1\) ways and fill the remaining 6 digits with odd numbers in \(5^6\) ways. Again we need to subtract the numbers having 0 in the first digit location
If first digit has a 0 , we can fill the remaining 6 digits with odd numbers in \(5^6\) ways.

Total no of 7 digit numbers whose sum of digits is even=
\(5^7-5^6\)+\(C^7_5*5^7-C^6_4*5^6\)+ \(C^7_3*5^7-C^6_2*5^6\)+\(C^7_1*5^7-5^6\)
=\(4*5^6+90*5^6+160*5^6+34*5^6\)
=\(288*5^6\)

Zareentaj - What's the OA ?

Bunuel/Ian
Can this approach be extended in general i.e., How many N digit integers are there whose sum is even/odd?
Answer = 9 x 10^(N-1)/2?

Kudos for the patience displayed. Although if taken this approach in GMAT I wonder if it can be done in 2 mins? Assume Bunuel or Ian's approach is what they would be looking for...

Wow! . Thanks Bunuel/IanStewart/crack700 for the explanation.

For \(n\geq{2}\) - yes.
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You can think of it in terms of subsequent number changes.

The first 7 digit number is 1,000,000. The sum of the digits will be 1, an odd number.
The next 7 digit number is 1,000,001. Its units digit will be 1 more than the previous so the sum of its digits will increase by 1 and hence be even.
and so on the sum will alternate between odd and even till after the next 8 numbers when the tens digit changes and we get 1,000,010. This sum is even and now again each subsequent number will increase by 1 and the sum of digits will alternate again between odd and even for the next 9 numbers. So 5 of them will be even.
And so on...

So exactly half the numbers will have even sum and half will have odd sum.

Total 7 digit numbers = 9 * 10*10*10*10*10*10 = 9,000,000
(The first digit cannot be 0. Rest all digits can take any value)

Half of them will have odd sum of digits. So 4,500,000
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Thanks for jotting down all the cases.. I had that approach in my mind but was too lazy to write it down on my paper.

Total 7 digit numbers = \(9*10^6\)
So half of them will have their digits sum even ,and other half will have their digits sum odd.
Answer for the question = Total 7 digit numbers/2 = \(4.5 * 10^6\) = 4500000

For the second step (5E2O) , why can't we write 4*5*5*5*5*5*5=4*(5)^6

First digit can be in 4 way (excluding 0) second digit can be 5 ways and so on...

KarishmaB
crack700
can you plz elaborate this issue...?

Posted from my mobile device

What if the first digit is not picked to be even? Then there are 5 ways in which it can be filled with an odd digit.
If the first digit is picked to be even, then there are 4 ways in which we can fill it.

Hence, a case like 1124242 is ignored when we do 4*5*5*5*5*5*5.


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