Question

if O represents number of integers between 10,000 and 100,000 all of whose digits are odd, and E represents number of integers between 10,000 and 100,000 all of whose digits are even, what is the value of O - E?

Answer 1

i)
fist let's find O, that is the number of all numbers between 10,000 and 100,000, whose all digits are odd.

so we have to count: 11,111; 11,113; ... 99,999

we notice that all these numbers are 5 digit, and they are made up of the odd digits {1, 3, 5, 7, 9} with repetition allowed.

so there are in total
`5*5*5*5*5= 5^(5)` such numbers.

ii)

E is the total number, of 5-digit numbers formed from the set{0, 2, 4, 6, 8} with repetition allowed, and the first digit not equal to 0.

so
`E=4*5*5*5*5= 4*5^(4)`


iii)
`O-E=5^(5)-4*5^(4)=5^(4)(5-4)=5^(4)=625`


Answer: 625