 # How many different letter arrangements can be made from the letters in the word MATHEMATICS? Lennie Carroll 2020-12-06 Answered
How many different letter arrangements can be made from the letters in the word MATHEMATICS?
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Step 1
Given:
The different letter arrangements are calculated from the word MATHEMATICS as follows,
Step 2
The formula is calculated as below,
$nPr=\frac{n!}{n1!n2!\dots nk!}$
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,
${n}_{1}\left(M\right)=2,{n}_{2}\left(A\right)=2,{n}_{3}\left(T\right)=2,{n}_{4}\left(H\right)=1,{n}_{5}\left(E\right)=1,{n}_{6}\left(I\right)=1,{n}_{7}\left(C\right)=1,{n}_{8}\left(S\right)=1$
Step 4
The input parameters are applied based on the nPr formula as follows,
$=\frac{11!}{2!2!2!1!1!1!1!1!}$
$=\frac{1×2×3×4×5×6×7×8×9×10×11}{\left(1×2\right)\left(1×2\right)\left(1×2\right)\left(1\right)\left(1\right)\left(1\right)\left(1\right)\left(1\right)}$
$=\frac{39916800}{8}$
Step 5
On dividing the value as,
$=\frac{39916800}{8}=4989600$
Hence, the letters of word "MATHEMATICS" 4989600 distinct ways of arrangement.

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