A museum curator is arranging 5 distinct marble statues and 2 distinct bronze statues in a… — GMAT

A museum curator is arranging $5$ distinct marble statues and $2$ distinct bronze statues in a single row along a gallery wall. If the $2$ bronze statues must not be placed next to each other, how many different linear arrangements of all $7$ statues are possible?

Cevap: 3600

Cevap

3,600

Using complementary counting, the total unrestricted arrangements of

7

distinct statues is

7! = 5,040

. The number of arrangements where the

2

bronze statues are placed together is determined by treating them as a single block:

6! \times 2! = 1,440

. Subtracting these forbidden arrangements from the total yields

5,040 - 1,440 = 3,600

valid linear arrangements.

Adım Adım Çözüm

Find total arrangements without restriction.

7! = 5,040

There are 7 distinct statues in total to arrange in a line.

Find arrangements where the 2 bronze statues are adjacent.

6! × 2! = 1,440

Grouping the 2 bronze statues into 1 block yields 6 items to order (6!), and the 2 bronze statues can swap positions inside the block (2!).

Apply complementary counting to find non-adjacent arrangements.

5,040 - 1,440 = 3,600

Subtracting the adjacent arrangements from total arrangements gives all valid arrangements.

Anahtar Kavram

Linear arrangements with non-adjacency restrictions using complementary counting.

Tahmini Süre:1m 30s

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