How many unique permutations are possible using the letters of the following words:

Question

Permutations 

2. How many unique permutations are possible using the letters of the following words:

a) ATHENS

b) BASKETBALL

c) ICICLE

d) SUBSTITUTE

Answer 1

a) ATHENS has 6 letters

6! =6x5x4x3x2x1 = 720 unique ways to arrange the letters

nPr = 6P1 = 6!/1! = 6!

b) BASKETBALL has 10 letters, but the permutations are less than 10! due to the 2 B's, 2A's and 2 L's

number of permutations = 453,600 < 10!

consider a more simple problem, with the word BLL. BLK has 3 letters and 3! = 6 permutations

But BLL has 2 L's

BLK

BKL

LBK

LKB

KLB

KBL are 6 ways =3!

but

BLL

LBL

LLB are just 3 ways if each L is identical = 3!/2! = 3

for BASKETBALL

10!/2!2!2! = 10!/8 =3628800/8 = 453,600

c) ICICLE is 6 letters with 2I's and 2C's

permutations = 6!/2!2! = 720/4 = 180

d) SUBSTITUTE is 10 letters with 2S's, 2U's and 3T's

permutations are 10!/2!2!3! = 10!/24 = 151,200

some relatively inexpensive hand calculators have permutation and combination functions