Question

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

Math Word Problem
ANSWERED
asked 2020-12-06
How many different letter arrangements can be made from the letters in the word MATHEMATICS?

Answers (1)

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.
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