Arrangements with Two People Not Together

“Not together” is a complement question: it’s far easier to count everything and subtract the “together” cases than to count “not together” directly. The “together” count uses the block method.

Five people are to be arranged in a row. In how many arrangements are two particular people, $A$ and $B$, not next to each other? [4]

The working

Step 1, count all arrangements of $5$ people: $5! = 120 \quad \text{(M1)}$

Step 2, count the “together” arrangements using the block method. Glue $A$ and $B$ into one unit, leaving $4$ units to arrange, then multiply by the $2!$ internal orders of the block: $4! \times 2! = 24 \times 2 = 48 \quad \text{(M1)}$

Step 3, subtract to get “not together”: $120 - 48 = 72 \quad \text{(M1, A1)}$

Where the marks are won and lost

• The complement is the efficient route: total $-$ together. Counting “not together” directly is far harder and error-prone.
• The “together” count needs the internal $2!$: the block $[AB]$ and $[BA]$ are different arrangements. Forgetting it gives $24$, halving the “together” count.
• Arrange $4$ units (block $+$ $3$ others), not $5$, once $A$ and $B$ are glued.

Common mistakes

• Forgetting the $\times 2!$ for the block’s internal order.
• Arranging $5$ units in the “together” case instead of $4$.
• Trying to count “not together” directly and missing cases.

Full method: Arrangements & Selections notes. Topic home: Permutations & Combinations pillar.