Total possibilities $=4! = 24.$
No. of ways 'Arun chooses red' or 'Shwetha chooses white'
$=\text{(no.of ways `Arun chooses red')}$
$+ \text{(no.of ways `Shwetha chooses white')}$
$-(\text{no.of ways `Arun chooses red' and `Shwetha chooses white')}$
$=6+6-2 = 10$
required $=24-10=14.$
Correct Answer: $D$