Question

For how many three digit numbers (100 to 999) is the sum of the digits even? (For example, 343 has an even sum of digits: 3 + 4 + 3 = 10 which is even.) Find the answer and explain why it is correct in at least two different ways.

Answer 1

there are 425 numbers of three digits whose the sum of the digits is even.

Explanation:

Option 1:

It is necessary to accomplish one of the next conditions:

(1) all of the digits are even.

(2) exactly 2 of the digits are odd

Then, for the first condition:

4x5x5 = 100 -> this is because the first digit can not be 0.

for the second condition:

(3C1)*5*5*5-1*5*5 =3*125-25=350

where 3C1 is a combination. The 3C1 determines which of the three digits is even, and it is necessary to subtract the numbers under 100 with the other 2 digits odd.

So, there are 450 numbers of three digits whose the sum of the digits is even.

Option 2:

there are 900 three digit number and 5*5*5 =125 has 3 odd numbers and 5*5*5+4*5*5+4*5*5=325 has 1 odd number, so there are 450 numbers of three digits whose the sum of the digitis is odd, so 900-450 whose the sum of the digits is even.