Question

The number of ways in which one can select three distinct integers between $1$ and $30$, both inclusive, whose sum is even, is

• A. $455$
• B. $1575$
• C. $1120$
• D. $2030$
• E. $1930$

Answer 1

The correct option is D

$2030$

Explanation for correct option:

Given that,
Number of integers $=$ $1$ to $30$
We need to select three distinct integers between $1$ to $30$, such that their sum is even. There are $15$ odd and $15$ even numbers.
Thus, numbers can be either all even or one even and other two odd.
Required number of ways $={}^{15}C_3+{}^{15}C_1\times {}^{15}C_2$
$$\begin{gathered} \therefore {}^{15}C_3+{}^{15}C_1\times {}^{15}C_2 \\ =\frac{15!}{3!\left(12!\right)}+\frac{15!}{1!14!}\times \frac{15!}{2!13!}\left[\because {}^{n}C_r=\frac{n!}{\left(r!\right)\left(n-r\right)!}\right] \\ =455+15\times 105 \\ =455+1575 \\ =2030 \end{gathered}$$
Hence, option D is correct.