JAMB 2019 Mathematics · Question 13

In how many ways can the word MATHEMATICIAN be arranged?

Options

A.6794800 ways
B.2664910 ways
C.6227020800 ways
D.129729600 ways

Answer: D. 129729600 ways

Explanation written with AI and checked against the answer key

To solve this problem, we use the formula for permutations of a multiset (arrangements of objects where some are identical).

1. Analyze the question

We need to find the number of distinct ways to arrange the letters in the word MATHEMATICIAN.

2. Explain the concept

When arranging letters, if some letters repeat, we use the formula:

\text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \dots \times n_k!}

Where:

$n$ is the total number of letters.
$n_1, n_2, \dots$ are the frequencies of the repeating letters.

3. Show the solution

Step 1: Count the total number of letters ( $n$ ). M-A-T-H-E-M-A-T-I-C-I-A-N Counting them: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. Total letters ( $n$ ) = 13

Step 2: Identify the repeating letters and their frequencies.

M: appears 2 times
A: appears 3 times
T: appears 2 times
H: appears 1 time
E: appears 1 time
I: appears 2 times
C: appears 1 time
N: appears 1 time

Step 3: Apply the formula.

\text{Arrangements} = \frac{13!}{2! \times 3! \times 2! \times 2!}

$13! = 6,227,020,800$
$2! = 2$
$3! = 6$

\text{Arrangements} = \frac{6,227,020,800}{2 \times 6 \times 2 \times 2}

\text{Arrangements} = \frac{6,227,020,800}{48}

\text{Arrangements} = 129,729,600

4. Identify the answer

Comparing our result to the given options, the correct value is 129,729,600. This matches option D.

5. Exam Tip

Always write out the letters and their counts clearly before plugging them into the formula. A common mistake is forgetting to count the "1s" (letters that appear only once), though they don't change the denominator (since $1! = 1$ ), it helps ensure you haven't missed any letters in your total count of 13!

AI can make mistakes. Report anything that looks wrong.