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Question

Difficulty: MediumPermutations and Linear Arrangements

A jewelry store window displays 66 distinct luxury watches—33 gold watches and 33 silver watches—in a single straight line from left to right. If no two gold watches can be placed adjacent to each other, how many different linear arrangements of the 66 watches are possible?

Answer: 144 arrangements

Answer

144
To place items such that no two gold watches are adjacent, we use the gap method. First, arrange the 3 distinct silver watches, which can be done in 3!=63! = 6 ways. Placing these 3 silver watches creates 4 potential spaces (one at each end and two between the silver watches). Next, select 3 of these 4 spaces to place the gold watches, which can be done in (43)=4\binom{4}{3} = 4 ways. Finally, arrange the 3 distinct gold watches within the selected spaces in 3!=63! = 6 ways. Multiplying these possibilities gives 6×4×6=1446 \times 4 \times 6 = 144 total distinct arrangements.

Step-by-Step Solution

1
Calculate the arrangements of the 3 distinct silver watches.
3! = 6 ways
The 3 silver watches are distinct and can be arranged among themselves in 3! ways.
2
Determine the available positions (gaps) for the gold watches.
4 available gaps
Placing 3 silver watches in a line creates 4 potential spaces (before the first, between adjacent pairs, and after the last) to ensure non-adjacency.
3
Select positions and arrange the 3 distinct gold watches.
C(4, 3) × 3! = 4 × 6 = 24 ways
Choosing 3 out of 4 gaps gives C(4, 3) = 4 combinations, and ordering the 3 distinct gold watches in those selected gaps gives 3! = 6 arrangements.
4
Multiply the possibilities from all independent steps using the Fundamental Counting Principle.
6 × 24 = 144
The total number of valid linear arrangements is the product of the number of ways to complete each step.

Key Concept

Linear Permutations with Non-Adjacent Restrictions (Gap Method)
Estimated Time:2m 0s
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Topics

Permutations and Linear Arrangements

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