IIT-JEE 1985Question 1
State TRUE or FALSE: If $n_1,n_2,\ldots,n_p$ are $p$ positive integers whose sum is an even number, then the number of odd integers among them is odd.
Permutation and Combination
Solution
Concept. The parity of a sum depends only on how many odd terms it has. Formulas used. (1) Sum of $t$ odd numbers is even $\iff t$ is evenStep 1 - Even terms. They do not change parity. Step 2 - Odd terms. If there are $t$ of them, the sum has the parity of $t$. Step 3 - Sum even. So $t$ is even, not odd. Example: $1+1=2$ has two odd terms. Step 4 - Verdict. FALSE. Answer: B