How many distinct arrangements are there for the letters in the word MATHEMATICIAN?

Question

ScanSolve AI Tutor · Accepted Answer

To find the number of ways the word MATHEMATICIAN can be arranged, we use the formula for permutations with repetitions.

Count the total number of letters in the word.
The word is MATHEMATICIAN.
Total number of letters, $n = 13$ .
Identify the repeated letters and their frequencies.
- M appears 2 times ( $n_M = 2$ )
- A appears 3 times ( $n_A = 3$ )
- T appears 2 times ( $n_T = 2$ )
- H appears 1 time
- E appears 1 time
- I appears 2 times ( $n_I = 2$ )
- C appears 1 time
- N appears 1 time
Apply the permutation formula for words with repeated letters.
The formula is:
$\frac{n!}{n_1! n_2! \cdots n_k!}$
where $n$ is the total number of letters, and $n_1!, n_2!, \ldots, n_k!$ are the factorials of the frequencies of the repeated letters.
Substitute the values and calculate.
$\text{Number of arrangements} = \frac{13!}{2! \cdot 3! \cdot 2! \cdot 2!}$
Calculate the factorials:
$13! = 6,227,020,800$
$2! = 2 \times 1 = 2$
$3! = 3 \times 2 \times 1 = 6$

Substitute these values into the formula:
$\text{Number of arrangements} = \frac{6,227,020,800}{2 \times 6 \times 2 \times 2}$ $\text{Number of arrangements} = \frac{6,227,020,800}{48}$ $\text{Number of arrangements} = 129,729,600$
Comparing this result with the given options:
A) 6794800 ways
B) 2664910 ways
C) 6227020800 ways
D) 129729600 ways

The calculated number of arrangements matches option D.

The final answer is $\boxed{\text{D) 129729600 ways}}$ .

Step-by-step solution