Chapter 3.8, Problem 26P

A salon offers three services-manicures, massages, and hair styling. Five clients come in for all three services. Forty clients come in for both hair styling and manicures. Thirty-five clients come in for both manicures and massages. Fifteen clients come in for both hair styling and massages. Ten clients come in only for massages. Five clients only come in for hair styling and only two clients come in for just manicures. How many total clients are represented by the data?

To determine

To calculate:How many total clients are represented by the data?

Answer to Problem 26P

97.

Explanation of Solution

Given information: 5 clients come for all three services.

40 clients come in for both hair styling and manicures.

35 clients come in for both manicures and massages.

15 clients come in for both styling and massages.

10 clients come in only for massages.

5 clients only come in for hair styling.

2 clients come in for just manicures.

Calculation: Let $x$ be the number of clients for massages , $y$ be the number of clients for hair styling , and $z$ be the number of clients for manicures.

using above data we plot the vein diagram

  

from the above vein diagram, we can say that

  $x\cap y\cap z=5$ and $x\cap y=15$

Now we can find the region that does not includes $z$ but includes $x\&y$

  $=15-5=10$

Now we find the region that does not includes $y$ but includes $x\&z$

  $=35-5=30$

finally ,we find the region that does not includes $x$ but includes $y\&z$

  $=40-5=35$

we know the principal of inclusion and exclusion

  $x\cup y\cup z=10+10+5++30+5+35+2=97$

Total number of clients are 97.