**RANKING AND SITTING ARRANGEMENT**

These section mainly deals with the positioning of a person or object when they are arranged in a row . Generally the position of a person from beginning and from end are given we need to find total number of person. $\\$ Lets understand this if position of Ram from beginning is 21 and from end is 11 then how we will find total number of persons. $\\$ Number of persons before Ram is 20 ( as he is $21^{st}$ from beginning hence excluding him is 20 ) and Number of persons behind him is 10 ( as he is 11 from end hence excluding him is 10)$\\$ Therefor Total number of persons are = Number of persons before Ram + Number of persons after Ram + Ram himself = 20 + 10 + 1 =31 $\\$ That is if rank of person from beginning is N and rank from end is M then total number of persons are ( N + M +1) $\\$ Some times total numbers of persons and rank of person from beginning / end is given And candidate is required to calculate rank from end / beginning. $\\$ Consider once again the above example now let’s say total number of persons are 31 is known and position of person is 11th from end then what will be his rank from beginning ? $\\$ Since there are 10 people behind him let us assume if these 10 people are removed from group then Ram is last person and Number of persons are 31 – 10 = 21 , Being the last in a row his rank from beginning will be 21 . $\\$ If N is total number of person and rank of person from end is M then his rank from beginning is ( N – M + 1 ) .$\\$ In another type of questions sitting arrangement of person around a circular table or in a row is to be determined . In which right and left neighbor is to be observed to make out the sitting of each person. $$ $$**EXAMPLE 1:-** In a class $\; 10^{th} \;$ a student Anil has $\;10^{th}\;$ position from top and $\;30^{th}\;$ position from bottom in an examination . How many students are there in class $\;10^{th}\;$ ? $$ $$ (A)$\;38\qquad\qquad\qquad (B)\;39\qquad\qquad\qquad (C)\; 40\qquad\qquad\qquad (D)\;41 $ $\\$ **ANSWER : –** 39 Total number of stents = ( Number of students before him ) + ( Number of students after him ) + (he himself ) $ = \;9\; +\;29\;+\;1\;=39 $ $$ $$**EXAMPLE 2:-**Thirty six vehicles are parked in a parking lot in a single row. After the first car there is one scooter , After the second car there are two scooters , After third car there are three scootors and so on . Find total number of scooters in a row ? $\\$ (A)$\;28\qquad\qquad\qquad (B)\;29\qquad\qquad\qquad (C)\; 30\qquad\qquad\qquad (D)\;31 $ $\\$ **ANSWER : –**28 $ \qquad$let C denote a car and S denote a scooter in parking lot. $\\$ In a Row they are arranged as CSCSSCSSSCSSSSCSSSSS………… $\\$ let total number of vehicles are $S_t$ $\\$ therefore $ S_t\;$ =(1 +1 ) + (1 + 2) + (1 + 3) +……+( 1 + n) + r $ \; where $ r = 0 ; 1 .$ S_t =\; n\; +\; \frac{n(n+1)}{2} \; +\; r \\ 36 = \frac {n(n+3)}{2} \; +\; r $ let r = 0 then $ \; n^2 + 3\,n-72\;=\;0;\qquad {n\notin{I}}$ $\\$ let r = 1 $\; n^2\;+\;3n\;-\;70\;=0\; \\ (n\;+\;10\,)(n\;-\;7)\;=\;0 \quad \Rightarrow n=7 \\$ Total number of scooters are $= \frac{7×8}{2}\;=\;28 $ $$ $$ **EXAMPLE 3:-** Five boys took part in a race. Aman finished before Dev But behind Ravi . Ram finished before Raj but behind Dev. Who was the winner of the race ? $\\$ (A)$\;Raj\quad\quad\qquad (B)\;Aman\quad\quad\qquad (C)\; Dev\quad\quad\qquad (D)\;Ravi $ $\\$**Answer :-** Ravi $\qquad$ 1. Ravi 2. Aman 3. Dev 4.Ram 5. Raj

**EXAMPLE 4** :- Four women$\;W_{1}\,,W_{2}\,, W_{3}\,,W_{4} \;$ and four men$\;M_{1}\,,M_{2}\,, M_{3}\,,M_{4} \;$ are sitting around a round table in the following way.$\\$ $\qquad W_{3}\,$ is between $\,M_{3}\;$ and to the right $\,M_{2}\;, \,W_{2}\;$ is to the right of $\,M_{1}\;,\,W_{4}\;$ is facing $\,W_{1}\;\,M_{3}\;$ is to the left of $\,W_{4}\;$ . No two men or women are sitting together . $\,M_{4}\;$is sitting between which two persons ? $\\$ (A)$\,W_{3}\;AND \;W_{4}\;$ (B)$\,W_{1}\; AND \;W_{3}\;$ ; (C)$\,W_{2}\; AND\;W_{4}\;$ (D)$\,W_{1}\; AND\;W_{2}\;$ $\\$ ** ANSWER :-** (D)