Question

In how many ways can 3 identical emeralds 2 identical diamonds and 2 different opals be arranged in a row in a display case

Answer 1

They can be arranged in 210 ways.

The arrangement of n objects where n₁ is one kind, n₂ is a second kind, and n₃ is a third kind is given by

$\frac{n!}{n_1!\times n_2!\times n_3!}$

There are 7 total gems; 3 of the first kind, 2 of the second kind and 2 of the third kind:

$\frac{7!}{3!2!2!} = 210$