Answer :
Answer:
The possible number of ways to select distinct (a, b) such that (a + b) is even is 534.
Step-by-step explanation:
The range 1 - 99 has 99 numbers, since 1 and 99 are inclusive.
Of these 50 numbers are odd and 49 are even.
The two distinct numbers a and b must have an even sum and a should be a multiple of 9.
The sum of two numbers is even only when both are odd or both are even.
The possible values that a can assume are,
a = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90 and 99}
Thus, a can assume 6 odd values and 5 even values.
- If a = odd number, then b can be any of the 49 out of 50 odd numbers.
Total number of ways to select a and b such that both are odd and their sum is even is:
[tex]n(Odd\ a\ and\ b)=n(Odd\ value\ of\ a)\times n(Odd\ value\ of\ b)=6\times49=294[/tex]
- If a = even number, then b can be any of the 48 out of 50 even numbers.
Total number of ways to select a and b such that both are even and their sum is even is
[tex]n(Even\ a\ and\ b)=n(E\ value\ of\ a)\times n(Even\ value\ of\ b)=5\times48=240[/tex]
Total number of ways to select distinct (a, b) such that (a + b) is even is =
[tex]=n(Odd\ a\ and\ b)+n(Even\ a\ and\ b)=294+240=534[/tex]
Thus, the possible number of ways to select distinct (a, b) such that (a + b) is even is 534.