Calculating the Number of Arrangements for the Word 'MATHEMATICS'

The word "Mathematics" is an eleven-letter word with specific repetitions of certain letters. In this article, we will walk through the method to calculate the number of arrangements for this word under no restrictions.

Understanding the Problem

The word "Mathematics" has the following letter frequencies:

M: 2 A: 2 T: 2 H: 1 E: 1 I: 1 C: 1 S: 1

Formula for Arrangements with Repeated Letters

The formula to calculate the number of arrangements of a word with repeated letters is given by:

Number of arrangements n! divide; (n_1! × n_2! × n_3! × ...)

where n is the total number of letters, and n_1, n_2, n_3, ... are the frequencies of the repeated letters.

Application to 'MATHEMATICS'

In the word "MATHEMATICS":

Total number of letters, n, is 11. Frequencies are: n_m 2, n_a 2, n_t 2, n_h 1, n_e 1, n_i 1, n_c 1, n_s 1.

Calculating the Number of Arrangements

To calculate the number of arrangements:

First, compute the factorial of the total number of letters, which is 11!. Then, compute the factorials of the frequencies of the repeated letters (M, A, T), which are 2! each. Use the formula: Number of arrangements 11! divide; (2! × 2! × 2! × 1! × 1! × 1! × 1! × 1!).

Let's break it down step-by-step:

11! 39,916,800. 2! 2 (for each M, A, T). (2! × 2! × 2!) 8. Therefore, Number of arrangements 39,916,800 div; 8 4,989,600.

Thus, the total number of arrangements of the letters in the word "MATHEMATICS" is 4,989,600.

Key Takeaways

Understanding the Formula: Use the formula to calculate arrangements when letters are repeated. Factorial Concept: Recall that 11! 11 × 10 × 9 × ... × 2 × 1, and so on for each factorial term. Application Process: Follow a systematic approach to calculate the number of arrangements using the given formula.

In conclusion, the word "MATHEMATICS" can be arranged in 4,989,600 different ways under no restrictions.