Chapter 1.9, Problem 13PSC

To determine

The different ways in which a diagram can be labeled.

Answer to Problem 13PSC

The different ways in which a diagram can be labeled are ABCD , ABDC , ADBC , ADCB ,ACBD , ACDB , DCBA , CDBA , BCDA ,CBDA , DBCA , BDCA ,DCAB , CDAB , DBAC , BDAC, BCAD , CBAD.

Explanation of Solution

Given information:

4 points are given in a plane.

Concept used:

  $Numberofwaystoarrangenthingsonndifferentplaces=n!$

Calculation:

For A to be collinear with two other points, A can be at 3 positions first , third and fourth.

and remaining 3 letter can be arrange in 3 positions in $3!$ .

  $\therefore Totalnumberofdifferentlebellingthatcanbedonewiththegivencondition=3\times 3!=18$

ABCD , ABDC , ADBC , ADCB ,ACBD , ACDB , DCBA , CDBA , BCDA ,CBDA , DBCA , BDCA ,DCAB , CDAB , DBAC , BDAC, BCAD , CBAD.

Conclusion:

Hence with given condition the diagram can be labeled in 18 different ways -

Few of these labeling are −