Question

Classify the problem according to whether it involves a permutation or a combination. In how many ways can the letters of the word GLACIER be arranged? combination permutation

Answer 1

The problem is an example of a permutation.
This is because, in a permutation, the ordering matters, so in this case, GLACIER and GLACIRE will be counted as two seperate permutations, but when we use combinations, the ordering is irrelevant, we are only checking for the distinct alaphabets and their selection(or arrangement), and the ordering (or order in which they are selected) is not considered

So, when we are considering how many ways the letters can be arranged, we have to use permutation

That is, we have to find out how many distinct ways there are to arrange 8 letters into 8 spots

The number of ways n distinct variables can be distinctly arranged into n spots is given by the formula

$^nP_n=n!/(n-n)!=n!/0!=n!/1=n!$

Substituting
00 =U
,we get
$^8P_8=8!/(8-8)!=8!/0!=8!/1=8!$

$8!=8*7*6*5*4*3*2*1=40320$

Hence, the letters of glaciers can be arranged in 40320 ways