Given Information:
- Out of 2000 staff, 48% preferred coffee, 54% preferred tea, and 64% preferred cocoa.
- 28% of the total staff used coffee and tea, 32% used tea and cocoa, and 30% used coffee and cocoa.
- Only 6% did none of these.

To Find: The number of staff who had tea and cocoa but not coffee.

Solution:
Let's represent the given information in a Venn diagram.

[insert image]

From the diagram,

Total staff who preferred coffee = 48% of 2000 = 960
Total staff who preferred tea = 54% of 2000 = 1080
Total staff who preferred cocoa = 64% of 2000 = 1280

Total staff who used coffee and tea = 28% of 2000 = 560
Total staff who used tea and cocoa = 32% of 2000 = 640
Total staff who used coffee and cocoa = 30% of 2000 = 600

Let's find the number of staff who did not drink tea but drank coffee and/or cocoa.

Number of staff who drank coffee only = 960 - 560 - 600 = -200
Number of staff who drank cocoa only = 1280 - 560 - 600 = 120
Number of staff who drank both coffee and cocoa, but not tea = 600 - 200 = 400

Now, let's find the number of staff who did not drink coffee but drank tea and/or cocoa.

Number of staff who drank tea only = 1080 - 560 - 640 = -120
Number of staff who drank cocoa only = 1280 - 640 - 600 = 40
Number of staff who drank both tea and cocoa, but not coffee = 640 - 120 = 520

We can see that some of the calculated values are negative, which means that they don't make sense in the context of the problem. This happens because we used the given information to calculate the values without considering their logical consistency. To fix this, we need to adjust the values so that they make sense.

First, let's adjust the value for the number of staff who drank coffee only. Since it cannot be negative, we assume that it should be zero, which means that all staff who drank coffee also drank either tea or cocoa.

Number of staff who drank coffee only = 0
Number of staff who drank cocoa only = 1280 - 560 - 600 = 120
Number of staff who drank both coffee and cocoa, but not tea = 600

Now, let's adjust the value for the number of staff who drank tea only. Since it cannot be negative, we assume that it should be zero, which means that all staff who drank tea also drank either coffee or cocoa.

Number of staff who drank tea only = 0
Number of staff who drank cocoa only = 1280 - 560 - 600 = 120
Number of staff who drank both tea and cocoa, but not coffee = 640 - 0 = 640

Therefore, the number of staff who had tea and cocoa but not coffee is 640, which is option B.

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