How many ways can you arrange 5 items if two have to be next to each other?

Answer 1

There are 48 possibilities where 2 of 5 items must be adjacent.

If you have 5 items, and they can be arranged in any order, there are 120

5 x 4 x 3 x 2 ways to arrange them, for example ABCDE, ABCED, and so forth.

However, if any two need to be placed next to each other, the number of variations is reduced to [4 x 3 x 2] x2 (=48), where there are only 4 separate "units" arranged, but the double-unit can appear with either of the pair first.

For example, if A and B must be together, you have 24 possibilities:

(AB)CDE

(AB)CED

(AB)DCE

(AB)DEC

(AB)EDC

(AB)ECD

C(AB)DE

C(AB)ED

D(AB)CE

D(AB)EC

E(AB)CD

E(AB)DC

CD(AB)E

DC(AB)E

CE(AB)D

EC(AB)D

DE(AB)C

ED(AB)C

CDE(AB)

CED(AB)

DEC(AB)

DCE(AB)

EDC(AB)

ECD(AB)

and another 24 where (AB) is replaced by (BA).