Before going to the answer lets recall some definitions to solve this problems.
Division Rule: If a finite set A is the pairwise disjoint subsets with d elements each ,then
$n=|A|/d$
Product Rule: If one event occur in $m$ ways and a second event occur in $n$ ways, the number of ways the two events occur in sequence is $m x n$ ways.
Sum Rule: If an event occur either in $m$ ways or in $n$ ways (non-overlapping) , the number of ways the event can occur is $m+n$ ways.
Subtraction Rule: If an event occur either in $m$ ways or in $n$ ways (overlapping) , the number of ways the event can occur is $m+n$ decreased by the number of ways the event can occur commonly to the two different ways.
Let A be the positive integers between the 100 and 999 inclusive. A contains 900 integers.
A. are divisible by 7?
Apply Division Rule:
We are interested in integers divisible by 7,
|A|=900, and d =7, substituting the values in Division Rule form ,
|900|/7 =128.5714 ≅ 128
Thus 128 integers are divisible by 7.
B.are odd?
An odd number is nothing but which is not divisible by 2.
By dividing with 2, we get Even numbers, By applying Division rule,
|900|/2=450
These 450 integers are even number. By subtracting these even integers from A we get the odd numbers i.e.,
900-450=450 (Odd intezers )
The Odd integers are 450.
C.have the same three decimal digits?
Since the sequence starts from 100 to 999,
we can't choose 0 in first place or in hundreds place. We have only 9 combinations (1 to 9).
since the same three decimal digits,the second place or tens must be same as first place, so we have only 1 combination.
since the same three decimal digits,the third place or units must be same as second place, so we have only 1 combination.
By using Product Rule:
9x1x1
Thus 9 integers have the same three decimal digits.
D.are not divisible by 4?
Check for integers divisible by 4 using product rule,
|900|/4 =225.( which are divisble by 4)
Now subtract it from total integers i.e.,
900-225=675
Thus 675 integers are not divisible by 4.
E.are divisible by 3 or 4?
Check for integers divisible by 3 using product rule,
|900|/3 =300
300 integers are divisible by 3.
225 integers are divisible by 4.(above proved)
Now integers divisible by both 3 and 4 thus divisible by 3 x4 =12 using product rule
|900|/12 = 75.
Number of Integers divisible by 3 or 4 are
using subtraction rule
Number of Integers divisible by 3 or 4 = integers divisible by 3 + integers divisible by 4 -integers divisible by both 3 and 4
= 300 + 225 - 75 = 450
Number of Integers divisible by 3 or 4 are 450.
F. are not divisible by either 3 or 4?
Number of integers which are not divisible by either 3 or 4 is
=Total Number of integers - Number of Integers divisible by 3 or 4
= 900 - 450 = 450
Number of integers which are not divisible by either 3 or 4 is 450
G. are divisible by 3 but not by 4?
Number of integers which are divisible by 3 but not by 4 is
= integers divisible by 3 - integers divisible by both 3 and 4
= 300 - 75 = 225
Number of integers which are divisible by 3 but not by 4 are 225.
H. are divisible by 3 and 4?
Integers divisible by both 3 and 4 thus divisible by 3 x4 =12 ,using product rule
|900|/12 = 75.
Thus 75 integers are divisible by 3 and 4.
