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## Here's the Solution to this Question

Let set A = consumers who took coffee

set B = consumers who took tea

set C = consumers who took cocoa

then

|A| = 230

|B| = 245

|C| = 325

|AC| = 70

|(A/B)/C| = 110

|(C/A)/B| = 185

|ABC| = 30

Two useful formulas:

EF + E/F = EF + EFc = E(F + Fc) = E, where Fc is complement set of F

EF + E/F = E (1)

using (1)

XZ + XY/Z + X/Y/Z = (XY + X/Y)Z + XY/Z + X/Y/Z = (XYZ +XY/Z) + (X/Y)Z +X/Y/Z =

XY +X/Y = X :

XZ + XY/Z + X/Y/Z = X (2)

using (1) again

AC/B + ABC = AC/B + ACB = AC: |AC/B| = |AC| - |ABC| = 70 - 30 =40 number of customers who took coffee and cocoa only

using (2): AC + AB/C + A/B/C = A: |AC| + |AB/C| + |A/B/C| = |A|,

|AB/C| = |A| - |AC| - |A/B/C| = 230 - 70 - 110 = 50 number of customers who took coffee and tea only

analogously:

|BC/A| = |C| - |C/A/B| - |AC| = 325 - 185 - 70 = 70 number of customers who took tea and cocoa only

using (1)

|AB| = |AB/C| + |ABC| = 50 + 30 = 80 number of customers who took coffee and tea

using (2)

|B/A/C| = |B| - |AB| - |BC/A| = 245 - 80 - 70 = 95 number of customers who took tea only

Number of customers who took tea coffee or cocoa:

|A/B/C| + |B/C/A| +|C/A/B| + |AB/C| + |AC/B| + |BC/A| + |ABC| =

110 + 95 + 185 + 50 + 40 + 70 + 30 = 580

Number of customers who took none of the beverages is 800 - 580 = 220

Answer: the number of customers who took tea only is 95,

the number of customers who took tea and cocoa only is 70,

the number of customers who took tea and coffee only is 50,

the number of customers who took none of the beverages is 220. 