Solutions
Number of people who like both apple and carrot juices are 50% more than the number of people who like all the three juices
So, the number of people who like both apple and carrot juices = 160 x 1.5 = 240
Number of people who like both apple and carrot juices but not orange juice = 240 – 160 = 80
Number of people who like both orange and carrot juice but not apple juice is twice the number of people who like both apple and carrot juice but not orange juice
Number of people who like both orange and carrot juice but not apple juice = 160
300 people like both apple and orange juices.
Number of people who like both apple and orange juice but not carrot juice = 300 – 160 = 140

Number of people who like only carrot juice and only apple juice are in the ratio of 5:8. Suppose their numbers are 5N and 8N respectively.
Number of people who like only orange juice is 25% less than the number of people who like only apple juice.
Number of people who like only orange juice = 6N
Total number of respondents = 5N+8N+6N+80+160+160+140 = 3200
=> 19N + 540 = 3200
=> N = 140

Required number = 160+80+160+140 = 540