Combinatorics

[marie]

A problem:

A jewelry store sells customized rings in which 3 gems selected by the customer are set in a straight row along the band of the ring. If exactly 5 different gems are available and if at least 2 gems in any given ring must be different, how many different rings are possible?

20
60
90
120
210

Please show all steps and reasoning,
Thxs!!

[cbenk121]

Breath Carolina - IDGAF
Five Finger Death Punch - Hard to See

Good problem, this had me stumped for a bit. At first I went down the combination route, but realized that was wrong.

So you have unlimited jewels essentially, so there's 5 possibilities for the first slot, 5 for the second, and 5 for the third slot.

5 x 5 x 5 = 125 = Number of possibilities.

However, at least two gems must be different. So we subtract out the possibilities where all the gems are the same. There are five types of gems, so there are five possibilities where all the gems would be the same.

125 - 5 = 120.

The answer is D.

Suppose there were four slots.

5 x 5 x 5 x 5 = 625.

Still only five possible scenarios where only one jewel would be used.

625 - 5 = 620.

Suppose there's three slots again, and all the slots had to be different?

5 x 4 x 3 = 60.

The reason I didn't do 5 x 5 x 4 to the original problem is that this cuts out possibilities. Suppose the one you cut out (by using 4 instead of 5) was Blue. You miss out on all these possibilities that could've been combined with Blue:

Red, Red
Red, Green
Red, Blue
Red, Yellow
Green, Red
etc..