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

$\begin{array}{1 1}(A)\;455\\(B)\;1575\\(C)\;1120\\(D)\;2030\end{array}$

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## 1 Answer

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Toolbox:
• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
Number are either all even or one even and other two odd.
Required number of ways =$15C_3+15C_1\times 15C_2$
$15C_3=\large\frac{15!}{3!12!}=\frac{15\times 14\times 13\times 12!}{3\times 2\times 12!}$
$\Rightarrow 455$
$15C_1=\large\frac{15!}{1!\times 14!}$
$\Rightarrow 15$
$15C_2=\large\frac{15!}{2!\times 13!}$
$\Rightarrow 105$
$15C_3+15C_1\times 15C_2=455+15\times 105$
$\Rightarrow 455+1575$
$\Rightarrow 2030$
Hence (D) is the correct answer.
answered Jun 20, 2014

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