What is the probability of choosing 2 distinct positive

BTGmoderatorDC wrote:What is the probability of choosing 2 distinct positive integers from the set of first 10 positive integers such that their product and sum are both even?

A. 1/5
B. 2/9
C. 2/5
D. 4/9
E. 1/2

OA B

Source: e-GMAT

Given: There is a list of first 10 positive integers: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

Note that the product of two integers is even if at least one between them is even. For example 2 and 3. Product of 2 and 3 = 2*3 = 6, even.

However, the sum of two integers is even if either both are even or both are odd. For example, 2 and 4. Sum of 2 and 4 = 2 + 4 = 6, even. Another example, 1 and 3. Sum of 1 and 3 = 1 + 3 = 4, even.

Thus, to maintain both the conditions that the sum, as well as the product, is even, we must have both the integers even.

# of ways of any choosing 2 even integers from {1, 2, 3, ..., 10} = 5C2 = 5.4/1.2 = 10; note that there 5 even integers

# of ways of any choosing 2 integers from {1, 2, 3, ..., 10} = 10C2 = 10.9/1.2 = 45;

Thus, the required probability = 10/45 = 2/9.

The correct answer: B

Hope this helps!

-Jay
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