Find the number of distinguishable permutations of the group of letters. $\mathrm { E } , \mathrm { S } , \mathrm { T } , \mathrm { I } , \mathrm { M } , \mathrm { A } , \mathrm { T } , \mathrm { E }$
A) 10,080
B) 8
C) 20,160
D) 40,320
E) 3360

Question

Quizplus · Accepted Answer

The total number of letters is 8, and there are duplicates: 2 T's and 2 E's. The formula for the number of distinguishable permutations is $ \frac { n! }{ n_1!n_2!...n_k! } $, where $ n $ is the total number of items, and $ n_1, n_2, ..., n_k $ are the counts of each duplicate item. Here, $ n = 8 $, $ n_1 = 2 $ (for T), and $ n_2 = 2 $ (for E), so the number of distinguishable permutations is $ \frac { 8! }{ 2!2! } = \frac { 40320 }{ 4 } = 10080 $.

Find the Number of Distinguishable Permutations of the Group of Letters