Permutation is the arrangement of n objects at n places which we know that it can be easily calculated as the 1st object can be arranged at one of the place in n ways , once one of the place is occupied for the 2nd we have only (n-1) places and it go on in the same way and leads to total number of ways as n! .
Here in we will see the arrangement of IDENTICAL OBJECTS with restriction that NO TWO of a particular identical object should be together.
let us assume we have n identical objects of one kind ( Say a ) and m identical objects of second kind ( Say b) , given n < m. What is the number of ways we can arrange them such that no two ‘a’ are together.
Here in we will first arrange all identical ” b ” at m places. The number ways is 1. ( Since all objects are identical hence interchange of places among b themselves will give us the same arrangement ).
Once we have arranged all b at m places now we go for arrangement of a i.e between b and including extremes we have (m+1) place of which we need to select n places where will place a. the number of ways will be $ {(m + 1)}\choose{n} $.
in the fig. there are 10 identical b and 6 identical a the number of arrangements are ${11}\choose{6} $
The same can be also used to select n persons sitting in a row from (n+m) such that no two of them are together. ( given n < m ).
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