A museum display curator is arranging 6 distinct historical coins—4 silver coins and 2 gol… — GRE General Test

A museum display curator is arranging $6$ distinct historical coins— $4$ silver coins and $2$ gold coins—in a single row inside a display case. If the $2$ gold coins cannot be placed adjacent to each other, how many different linear arrangements of the $6$ coins are possible?

A
$240$
B
$360$
$480$ Cevap
D
$576$
E
$720$

Cevap

The total number of valid linear arrangements is

480

Arranging the

4

distinct silver coins yields

4! = 24

orderings. These

4

coins create

5

available gaps (including both ends). To guarantee the

2

distinct gold coins are not adjacent, each gold coin must occupy a separate gap. The number of ways to assign

2

distinct gold coins to

5

gaps is

P(5, 2) = 5 \times 4 = 20

. By the Fundamental Counting Principle, the total number of valid linear arrangements is

24 \times 20 = 480

Adım Adım Çözüm

Calculate the total number of ways to arrange the unrestricted silver coins.

The

4

distinct silver coins can be arranged in

4! = 24

ways.

The relative positions of the silver coins matter because each coin is distinct.

Determine the number of available gap positions created by the silver coins to separate the gold coins.

Arranging

4

silver coins creates

5

possible gaps (one before the first coin, three between adjacent silver coins, and one after the last coin: \_ S \_ S \_ S \_ S \_).

Placing at most one gold coin per gap guarantees that the two gold coins will not be adjacent.

Calculate the number of ways to place the

2

distinct gold coins into the

5

available gaps.

The number of ways to place the

2

distinct gold coins into

5

distinct slots is

P(5, 2) = 5 \times 4 = 20

Order matters because the gold coins are distinct objects.

Apply the Fundamental Counting Principle to find the total arrangements.

Total arrangements =

24 \times 20 = 480

The arrangement of the silver coins and the placement of the gold coins are independent sequential choices.

Anahtar Kavram

Slot method for non-adjacent arrangements using permutations and the Fundamental Counting Principle

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