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A certain firm has 28 lawyers in three states. These lawyers have lice

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Bunuel
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A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   22 Jan 2022, 11:24
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A certain firm has 28 lawyers in three states. These lawyers have license either in only one state or all three states. If 10 have license in state X, 11 have license in Y, 13 have license in Z. How many lawyers have license only in state Z ?

A. 3
B. 6
C. 10
D. 11
E. 13
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C
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yashikaaggarwal
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A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   22 Jan 2022, 12:43
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Total => 28 (A+B+C+D) ----------(1)
These lawyers have license either in only one state or all three states. If 10 have license in state X, 11 have license in Y, 13 have license in Z.
A+D = 10
B+D = 11
C+D = 13
Adding all 3
A+B+C+3D = 34 ------------(2)

Subtracting equation 1 from equation 2
A+B+C+3D = 34
A+B+C+D = 28
----------------
2D = 6 => D = 3

Now to get only C
C+D = 13
C = 13-3 => 10

Answer is C

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chetan2u
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   22 Sep 2023, 19:05
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Bunuel wrote:
A certain firm has 28 lawyers in three states. These lawyers have license either in only one state or all three states. If 10 have license in state X, 11 have license in Y, 13 have license in Z. How many lawyers have license only in state Z ?

A. 3
B. 6
C. 10
D. 11
E. 13



X = Only x + All
Y = Only y + All
Z = Only z + All
X+Y+Z=Only x + All +Only y + All + Only z + All
10+11+13=(Only x +Only y + Only z + All) + 2*All
34=28+2*All
All=3
So only z=Z-All=13-3=10
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   22 Sep 2023, 22:48
Bunuel wrote:
A certain firm has 28 lawyers in three states. These lawyers have license either in only one state or all three states. If 10 have license in state X, 11 have license in Y, 13 have license in Z. How many lawyers have license only in state Z ?

A. 3
B. 6
C. 10
D. 11
E. 13


X = 10
Y = 11
Z = 13

X U Y U Z = X + Y + Z - X ∩ Y - Y ∩ Z - Z ∩ X + X ∩ Y ∩ Z

Now no two cities share lawyers, hence X ∩ Y - Y ∩ Z - Z ∩ X = 0

As a result of this scenario, we'll have to tweak the original formula and subtract the 2 * X ∩ Y ∩ Z as the segments don't overlap between 2 cities hence, we cannot subtract twice the tri-overlap area.

Hence X U Y U Z = X + Y + Z - 0 - 2 * X ∩ Y ∩ Z
=> 28 = 10 + 11+ 13 - 2 * X ∩ Y ∩ Z
=> 6 = 2 * X ∩ Y ∩ Z
=> X ∩ Y ∩ Z = 3

In order to find lawyers working only in City Z, we have to subtract lawyers working in all 3 cities from the total lawyers from City Z

Z - X ∩ Y ∩ Z
=> 13 - 3 = 10

Option C
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   06 Jan 2024, 02:44
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Bunuel, this is a GMAT Prep Focus mock question, please tag it.
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Bunuel
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   06 Jan 2024, 06:08
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AnkurGMAT20 wrote:
Bunuel, this is a GMAT Prep Focus mock question, please tag it.

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Added the tag. Thank you!
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   31 Mar 2024, 01:02
Can someone explain why the formula "Total=A+B+C−(AnB+AnC+BnC)+AnBnC+Neither" cannot be used to solve this question?
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Bunuel
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   31 Mar 2024, 02:42
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zaprico wrote:
A certain firm has 28 lawyers in three states. These lawyers have license either in only one state or all three states. If 10 have license in state X, 11 have license in Y, 13 have license in Z. How many lawyers have license only in state Z ?

A. 3
B. 6
C. 10
D. 11
E. 13

Can someone explain why the formula "Total=A+B+C−(AnB+AnC+BnC)+AnBnC+Neither" cannot be used to solve this question?

Because each of AnB, AnC, and BnC also includes AnBnC, and thus the sum of AnB + AnC + BnC won't be 0. You should use the second formula for three overlapping sets:

\(Total = A + B + C - (sum \ of \ EXACTLY \ 2-group \ overlaps) - 2*(all \ three) + Neither\).

28 = 10 + 11 + 13 - 0 - 2*(all three) + 0

(all three) = 3

Only Z = Z - (all three) = 13 - 3 = 10.

Answer: C.

For further insights into the distinctions between these two formulas, check out ADVANCED OVERLAPPING SETS PROBLEMS.

Hope it helps.­
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink] New post   04 Apr 2024, 02:58
zaprico wrote:
Can someone explain why the formula "Total=A+B+C−(AnB+AnC+BnC)+AnBnC+Neither" cannot be used to solve this question?

­Because it's either 1 or all 3 --> there's no option of 2 which the formula includes
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Re: A certain firm has 28 lawyers in three states. These lawyers have lice [#permalink]
04 Apr 2024, 02:58
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