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Question

Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.

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Solution

Let X denote the set of all 3-digit natural numbers
Let O have only odd digits and E have only even digits
Then X (OE) is the set of all 3-digit natural numbers having atleast one odd digit and atleast one even digit. The desired sum is therefore
xXx=yYyzZz
xXx=999j=1j99k=1k
=999×1000299×1002
=50×9891=494550
Consider O=(1,3,5,7,9). Suppose the digit in unit's place is 1. We can fill the digit in ten's place in 5 ways and the digit in hundred's place in 5 ways. Thus there are 25 numbers in the set O each of which has 1 in its unit's place. Similarly, there are 25 numbers whose digit in unit's place is 3; 25 having its digit in unit's place as 5; 25 with 7 and 25 with 9.
Thus the sum of the digits in unit's place of all the numbers in O is
25(1+3+5+7+9)=25×25=625
A similar argument shows that the sum of digits in ten's place of all the numbers in O is 625 and that in hundred's place is also 625. thus the sum of all the numbers in O is
625(102+10+1)=625×111=69375
Consider E=(0,2,4,6,8).
The digit in hundred's place is never 0. Suppose the digit in unit's place is 0. There are 4×5=20 such numbers. Similarly, 20 numbers each having digits 2,4,6,8 in their unit's place. Thus the sum of the digits in unit's place of all the numbers in E is
20(0+2+4+6+8)=20×20=400
A similar reasoning shows that the sum of the digits in ten's place of all the numbers in E is 400, but the sum of the digits in hundred's place of all the numbers in E is 25×25=500. Thus the sum of all the numbers in E is
500×102+400×10+400=54400.
The required sum is
4945506937554400=370775

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