A panel of 150 sommeliers evaluated three vintages of wine: Cabernet, Pinot Noir, and Syra… — GMAT

A panel of $150$ sommeliers evaluated three vintages of wine: Cabernet, Pinot Noir, and Syrah. Every sommelier rated at least one vintage as Exceptional. Overall, $79$ sommeliers rated Cabernet as Exceptional, $71$ rated Pinot Noir as Exceptional, and $84$ rated Syrah as Exceptional. Exactly $18$ sommeliers rated all three vintages as Exceptional, and exactly $27$ rated ONLY Cabernet as Exceptional. If the total number of sommeliers who rated BOTH Cabernet and Pinot Noir as Exceptional is twice the number of sommeliers who rated ONLY Pinot Noir and Syrah as Exceptional, how many sommeliers rated ONLY Cabernet and Syrah as Exceptional?

Cevap: 24

Cevap

Applying the 3-set Inclusion-Exclusion Principle determines that

48

sommeliers rated exactly two vintages. Combining the Cabernet set total (

79

) with the given ONLY Cabernet count (

27

) establishes that

cp + cs = 34

. Substituting

cp = 2ps - 18

into these two relationships forms a system of linear equations (

3ps + cs = 66

and

2ps + cs = 52

). Solving this system yields

ps = 14

and

cs = 24

sommeliers who rated ONLY Cabernet and Syrah as Exceptional.

Adım Adım Çözüm

Calculate the sum of all pairwise set overlaps using the Inclusion-Exclusion Principle.

|C \cap P| + |P \cap S| + |C \cap S| = 102

The total union is equal to the sum of individual set sizes minus the sum of pairwise intersections plus the three-set intersection.

Determine the sum of regions representing sommeliers who rated exactly two vintages.

cp + ps + cs = 48

Each pairwise intersection consists of an 'exactly two' region plus the 'all three' region (

18

). Subtracting

3 \times 18 = 54

from

102

leaves

48

Set up linear equations using the given ratio and Cabernet set total.

3ps + cs = 66

and

2ps + cs = 52

Expressing

cp = 2ps - 18

and substituting it into

cp + ps + cs = 48

gives the first equation; substituting into the Cabernet total

27 + cp + cs + 18 = 79

gives the second equation.

Solve the system of equations for the target region

cs

ps = 14

and

cs = 24

Subtracting the two equations yields

ps = 14

, and substituting back gives

cs = 24

Anahtar Kavram

Three-set Venn diagram algebraic modeling and inclusion-exclusion principle

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