Six distinct letters—A, B, C, D, E, and F—are to be arranged in a single line. How many di… — GMAT

Six distinct letters— $A, B, C, D, E,$ and $F$ —are to be arranged in a single line. How many different linear arrangements are possible such that letter $A$ appears somewhere to the left of letter $B$ , and letters $C$ and $D$ are not adjacent to each other?

Answer: 240

Answer

240

To find the number of valid linear arrangements, apply symmetry and the complement rule. First, in half of all

6! = 720

arrangements (360 arrangements), letter A appears to the left of letter B. Next, find the number of arrangements where A is to the left of B AND letters C and D are adjacent. Treating C and D as one combined block gives

5! \times 2! = 240

total arrangements where C and D are adjacent. By symmetry, letter A appears to the left of letter B in half of these cases (

\frac{240}{2} = 120

). Subtracting these 120 restricted arrangements from the 360 total arrangements where A precedes B gives

360 - 120 = 240

Step-by-Step Solution

Determine the number of linear arrangements in which letter A appears somewhere to the left of letter B.

360 arrangements

For 6 distinct letters, there are

6! = 720

total linear arrangements. By symmetry, letter A appears before letter B in exactly half of all arrangements:

\frac{720}{2} = 360

Determine the number of arrangements where letter A is to the left of letter B AND letters C and D are adjacent.

120 arrangements

Treating C and D as a single block yields 5 items to arrange, which can be done in

5! = 120

ways. The block itself has

2! = 2

internal orderings, giving

120 \times 2 = 240

arrangements where C and D are adjacent. By symmetry, letter A appears before letter B in half of these arrangements:

\frac{240}{2} = 120

Subtract the arrangements where C and D are adjacent from the total arrangements where A is to the left of B.

240 arrangements

The number of arrangements where letter A is to the left of letter B and letters C and D are not adjacent is

360 - 120 = 240

Key Concept

Permutations with Relative Position and Non-Adjacency Restrictions

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