Given: $\hat i \times (\hat j + \hat k)+ $$\hat j \times (\hat k + \hat i)+ $$\hat k \times (\hat i + \hat j)$
$\hat i \times (\hat j + \hat k) = \hat i \times \hat j + \hat i \times \hat k$
$\hat j \times (\hat k + \hat i) = \hat j \times \hat k+ \hat j \times \hat i$
$\hat k \times (\hat i + \hat j) = \hat k \times \hat i + \hat k \times \hat i$
$\Rightarrow \hat i \times \hat j + \hat i \times \hat k + \hat j \times \hat k+ \hat j \times \hat i + \hat k \times \hat j + \hat k \times \hat i$
Given that $\hat i \times \hat j = - \hat j \times \hat i$:
$\Rightarrow \hat i \times \hat j + \hat i \times \hat k + \hat j \times \hat k -\hat i \times \hat j - \hat j \times \hat k - \hat i \times \hat k$
$\Rightarrow = 0$