document.write( "Question 1017311: Consider set of first 18 natural numbers. Five different
\n" ); document.write( "numbers a1,a2,a3,a4,a5 are to be selected from set S under
\n" ); document.write( "the condition that a(i +1) -a(i)≥2 for all i= 1,2,3,4.
\n" ); document.write( "Find the number of ways (T) in which selection of 5 numbers
\n" ); document.write( "can be done based on the given condition.
\n" ); document.write( "Options are:
\n" ); document.write( "a- 50≤T≤300
\n" ); document.write( "b- 250≤T≤800
\n" ); document.write( "c-700≤T≤2500
\n" ); document.write( "d- 2000≤T≤3500
\n" ); document.write( "Please provide the solution. So that I can solve other
\n" ); document.write( "related problems.
\n" ); document.write( "Thanks \n" ); document.write( "
Algebra.Com's Answer #634141 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "This asks for the number of ways to choose 5 numbers from\r\n" );
document.write( "\r\n" );
document.write( "1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18\r\n" );
document.write( "\r\n" );
document.write( "such that no two of the 5 are next to each other:\r\n" );
document.write( "\r\n" );
document.write( "For illustration, suppose the 5 numbers chosen were\r\n" );
document.write( "3,7,12,14,16.  Let's bracket those 5:\r\n" );
document.write( "\r\n" );
document.write( "1,2,[3],4,5,6,[7],8,9,10,11,[12],13,[14],15,[16],17,18\r\n" );
document.write( "\r\n" );
document.write( "For all but the last one, 16, let's bring the right-hand \r\n" );
document.write( "neighbor into the brackets:\r\n" );
document.write( "\r\n" );
document.write( "1,2,[3,4],5,6,[7,8],9,10,11,[12,13],[14,15],[16],17,18\r\n" );
document.write( "\r\n" );
document.write( "This is a sequence of 14 things, 5 brackets containing\r\n" );
document.write( "numbers and 9 numbers not within brackets.  And 5 of the\r\n" );
document.write( "14 indicate 5 choices according to the required condition\r\n" );
document.write( "of no two chosen are next to each other.\r\n" );
document.write( "\r\n" );
document.write( "Here is another example which shows why it is not necessary\r\n" );
document.write( "to bring in a neighbor to the right of the largest number\r\n" );
document.write( "chosen into a bracket.\r\n" );
document.write( "\r\n" );
document.write( "Suppose the 5 numbers chosen were 1,5,10,12,18.  Let's bracket \r\n" );
document.write( "those 5:\r\n" );
document.write( "\r\n" );
document.write( "[1],2,3,4,[5],6,7,8,9,[10],11,[12],13,14,15,16,17,[18]\r\n" );
document.write( "\r\n" );
document.write( "For all but the last one, 18, let's bring the right-hand \r\n" );
document.write( "neighbor into the bracket:\r\n" );
document.write( "\r\n" );
document.write( "[1,2],3,4,[5,6],7,8,9,[10,11],[12,13],14,15,16,17,[18]\r\n" );
document.write( "\r\n" );
document.write( "This shows why we don't need to bring a right hand neighbor\r\n" );
document.write( "into the last number chosen, because in this case there is\r\n" );
document.write( "no right hand neighbor.  It is necessary that only the \r\n" );
document.write( "first four numbers chosen have a right hand neighbor, which\r\n" );
document.write( "of course, must not have not been chosen as one of the 5. \r\n" );
document.write( "\r\n" );
document.write( "Notice that this is also a sequence of 14 \"things\", consisting\r\n" );
document.write( "of 5 brackets containing numbers and 9 numbers not within \r\n" );
document.write( "brackets.  And 5 of these 14 indicate 5 choices according to \r\n" );
document.write( "the required condition that no two chosen are next to each other.\r\n" );
document.write( "\r\n" );
document.write( "So the number of such choices is the number of ways to choose\r\n" );
document.write( "5 positions from the 14 possible positions to place the 5 brackets.\r\n" );
document.write( "\r\n" );
document.write( "Answer: 14C5 = 2002\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );