Question

Two different integers are selected at random from the integers $1$ to $12$ inclusive. What is the probability that the sum of the two numbers is even?

• A. $\frac{1}{18}$
• B. $\frac{1}{2}$
• C. $\frac{5}{9}$
• D. $\frac{4}{11}$
• E. $\frac{5}{11}$

Answer 1

The correct option is E $\frac{5}{11}$

From $12$ numbers, $2$ numbers can be selected in ${{}^{12}C}_2=66$ ways.
If the two numbers are even then the sum is odd, so number of ways is ${{}^{6}C}_2=15$
If the two numbers are odd then the sum is even, so number of ways is ${{}^{6}C}_2=15$
So, the total number of ways, we can select $2$ numbers such that the sum is even is $30$
The probability is $\frac{30}{66}=\frac{5}{11}$