Question

# How many different letter arrangements can be made from the letters in the word MATHEMATICS?

Math Word Problem
How many different letter arrangements can be made from the letters in the word MATHEMATICS?

2020-12-07
Step 1
Given:
The different letter arrangements are calculated from the word MATHEMATICS as follows,
Step 2
The formula is calculated as below,
$$\displaystyle{n}{P}{r}=\frac{{{n}!}}{{{n}{1}!{n}{2}!\ldots{n}{k}!}}$$
Step 3
The value of the input parameters and values as follows,
The alphabets having the total number is n and the subset follows the values as (n1, n2, . . nk) based on the word "MATHEMATICS" with the value of n=11
The subsets are as follows,
Subsets : M = 2, A = 2, T = 2, H = 1, E = 1, I = 1, C = 1, S = 1,
$$\displaystyle{n}_{{1}}{\left({M}\right)}={2},{n}_{{2}}{\left({A}\right)}={2},{n}_{{3}}{\left({T}\right)}={2},{n}_{{4}}{\left({H}\right)}={1},{n}_{{5}}{\left({E}\right)}={1},{n}_{{6}}{\left({I}\right)}={1},{n}_{{7}}{\left({C}\right)}={1},{n}_{{8}}{\left({S}\right)}={1}$$
Step 4
The input parameters are applied based on the nPr formula as follows,
$$\displaystyle=\frac{{{11}!}}{{{2}!{2}!{2}!{1}!{1}!{1}!{1}!{1}!}}$$
$$\displaystyle=\frac{{{1}\times{2}\times{3}\times{4}\times{5}\times{6}\times{7}\times{8}\times{9}\times{10}\times{11}}}{{{\left({1}\times{2}\right)}{\left({1}\times{2}\right)}{\left({1}\times{2}\right)}{\left({1}\right)}{\left({1}\right)}{\left({1}\right)}{\left({1}\right)}{\left({1}\right)}}}$$
$$\displaystyle=\frac{39916800}{{8}}$$
Step 5
On dividing the value as,
$$\displaystyle=\frac{39916800}{{8}}={4989600}$$
Hence, the letters of word "MATHEMATICS" 4989600 distinct ways of arrangement.