document.write( "Question 1195171: 3.
\n" ); document.write( " Happy Harold’s Autos collected data on cars sold in October.
\n" ); document.write( "43 Total number of cars sold
\n" ); document.write( "23 Red cars
\n" ); document.write( "25 Cars with Air Conditioning
\n" ); document.write( "29 Cars with Heated Seats
\n" ); document.write( "17 Red cars with Air Conditioning
\n" ); document.write( "15 Red cars with Heated Seats
\n" ); document.write( "16 Cars with Air Conditioning and Heated Seats
\n" ); document.write( "10 Red cars with Air Conditioning and Heated Seats\r
\n" ); document.write( "\n" ); document.write( "(a) Draw a Venn diagram illustrating all of the above information.
\n" ); document.write( "(b) How many red cars had neither Air Conditioning nor Heated Seats?
\n" ); document.write( "(c) How many cars had Airconditioning were not red?
\n" ); document.write( "(d) How many cars were not red and had neither Air Conditioning nor Heated Seats?
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #827569 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "You are given the universal set U of 43 cars and 3 its basic subsets R, A and H \r\n" );
document.write( "(see the table below).\r\n" );
document.write( "\r\n" );
document.write( "    U    43    total cars\r\n" );
document.write( "\r\n" );
document.write( "    R    23    Red cars\r\n" );
document.write( "\r\n" );
document.write( "    A    25    cars with Air conditioning\r\n" );
document.write( "\r\n" );
document.write( "    H    29    cars with Heated seats\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Also, you are given info about their in-pair intersections and about their triple intersection.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    RA   17    Red cars with Air conditioning\r\n" );
document.write( "\r\n" );
document.write( "    RH   15    Red cars with Heated seats\r\n" );
document.write( "\r\n" );
document.write( "    AH   16    cars with Air conditioning and Heated seats\r\n" );
document.write( "\r\n" );
document.write( "    RAH  10    Red cars with Air conditioning and Heated seats\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Having this info well organized, you can easily answer all questions (b), (c) and (d) \r\n" );
document.write( "without drawing Venn diagram.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(b)  The set {Red cars had neither Air Conditioning nor Heated Seats} is  R \ ((RA U RH) \ RAH)   \r\n" );
document.write( "\r\n" );
document.write( "     and their number is  23 - (17 + 15 - 10)  = 1.    ANSWER                                     \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(c)  The set {cars had Air Conditioning were not red} is  A \ RA,\r\n" );
document.write( "\r\n" );
document.write( "     and their number is  25 - 17 = 8.     ANSWER\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(d)  The set {cars were not red and had neither Air Conditioning nor Heated Seats} is  U \ (R U A U H).\r\n" );
document.write( "     \r\n" );
document.write( "     So, calculate the number of cars in the union (R U A U H}  first.\r\n" );
document.write( "\r\n" );
document.write( "     For it, use the inclusion-exclusion princuple/(formula)\r\n" );
document.write( "\r\n" );
document.write( "         n(R U A U H) = n(R) + n(A) + n(H) - n(RA) - n(RH) - n(AH) + n(RAH) = \r\n" );
document.write( "\r\n" );
document.write( "                      =  23  +  25  +  29  -  17   -  15   -  16   +  10    = 39.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Now the last step gives the answer to question (d) :  \r\n" );
document.write( "\r\n" );
document.write( "        the number of cars were not red and had neither Air Conditioning nor Heated Seats = 43 - 39 = 4.    ANSWER\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On inclusion-exclusion principle,  see this Wikipedia article\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To see many other similar  (and different)  solved problems,  see the lessons\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "    - Counting elements in sub-sets of a given finite set\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problems on counting elements in sub-sets of a given finite set\r
\n" ); document.write( "\n" ); document.write( "    - Challenging problems on counting elements in subsets of a given finite set \r
\n" ); document.write( "\n" ); document.write( "    - Selected problems on counting elements in subsets of a given finite set \r
\n" ); document.write( "\n" ); document.write( "    - Inclusion-Exclusion principle problems \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Happy learning (!)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );