Question

bob is about to hang his 8 shirts in the wardrobe. he has four different styles of shirt two identical ones of each particular style. how many different arrangements are possible if no two identical shirts are next to one another? a

Answer 1

4 arrangements are possible

Answer 2

Answer:

Number of ways with no identical shirts next to one another is 2520−1656= 864 ways

Step-by-step explanation:

Total number of shirts = 8

Number of shirts with identical pairs can be arranged in $\frac{8!}{(2!)^{4} }$ ways = 2520

To find the number of ways in which no identical two shirts are hanged together we subtract the number of ways in which  4 pairs are together, 3 pairs are together, 2 pairs together and 1 pair together.

Let the shirts be called A, A, B, B, C, C, D, D respectively.

(i)  Imagine each pair as one object.

no. of ways in which s{A,B,C,D} can be arranged  =4! = 24 ways

There are $\binom{4}{4}$  subsets of type  i e {A,B,C,D}

Total number of 8-shirts of this type is 24.

(ii) s{A,B,C} = 36. To see this we first permute the 5 objects AA, BB, CC, D, D.

No. of ways = $\frac{5!}{2!}$ =60 ways but includes {A,B,C,D}. Hence s{A,B,C} =  60−24  = 36 ways

There are $\binom{4}{3}$ subsets i.e {A,B,C} {A,B,D} {A,C,D} {B,C,D}  

Total number of 8-shirts of this type is 4×36=144

(iii)  No. ways =  6!/(2!)^2 way = 180 ways but it includes {A,B,C}, {A,B,D} and {A,B,C,D}.

Hence s{A,B} =  180- 2×36 -24 = 84

There are $\binom{4}{2}$ subsets of type i e {A,B} {A,C} {A,D} {B,C} {B,D} {C,D}

Total number of 8-shirts of this type is 6×84=504

(iv) permuting the 7 objects AA, B, B, C, C, D, D = $\frac{7!}{2!^{3} }$

                                                                              = 630  ways

Then removing from it the subsets {A,B}, {A,C}, {A,D}, {A,B,C}, {A,B,D}, {A,C,D} and {A,B,C,D}. Hence s{A}=  630−3×36−3×84–24 = 246

Finally there are $\binom{4}{4}$ subsets of type ie {A} {B} {C} {D}

Total number of 8-shirts of this type=4×246=984

Therefore, number of ways with some identical shirts been together is 24+144+504+984=1656

Number of ways with no identical shirts next to one another is 2520−1656= 864 ways