Explained: "How many different letter arrangements can be made..."

High School
Math Word Problem

Lennie Carroll

Answered question

2020-12-06

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

Answer & Explanation

rogreenhoxa8

Skilled2020-12-07Added 109 answers

Step 1
Given:
The different letter arrangements are calculated from the word MATHEMATICS as follows,
Step 2
The formula is calculated as below,

n P r = \frac{n!}{n 1! n 2! \dots 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,

n_{1} (M) = 2, n_{2} (A) = 2, n_{3} (T) = 2, n_{4} (H) = 1, n_{5} (E) = 1, n_{6} (I) = 1, n_{7} (C) = 1, n_{8} (S) = 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 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11}{(1 \times 2) (1 \times 2) (1 \times 2) (1) (1) (1) (1) (1)}

= \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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