https://file.aiquickdraw.com/mathbot/file/17264765540951pg0z7o9.jpg

The problem in the image asks:

**"How many words can be formed by taking 4 letters at a time out of the letters of the word 'MATHEMATICS'?"**

Let's break it down.

The word "MATHEMATICS" consists of 11 letters, but some letters are repeated. Specifically:
- M: 2 times
- A: 2 times
- T: 2 times
- H, E, I, C, S: 1 time each

We need to calculate how many distinct combinations of 4-letter words can be formed using these letters, accounting for repetitions.

This is a combinatorics problem with repeated elements, so the calculation involves the concept of combinations of multisets.

Let me calculate this step by step for you.The total number of distinct 4-letter words that can be formed from the letters of the word "MATHEMATICS," accounting for repeated letters, is **157**.

Would you like to explore further details on this calculation, or do you have any questions?

Here are 5 related questions that expand on this topic:
1. How would the result change if we formed 5-letter words instead of 4?
2. How many words can be formed using all the letters of "MATHEMATICS"?
3. What would the calculation look like if no letter repetitions were allowed?
4. How can permutations be used to arrange the letters differently?
5. What is the probability of randomly picking 4 letters and getting all different letters?

**Tip**: When working with repeated elements in combinatorics, consider each case for how many repetitions are possible.

How many words can be formed by taking 4 letters at a time out of the letters of the word 'MATHEMATICS'?

Explore how many four-letter combinations can be formed from the word 'MATHEMATICS', considering repeated letters. This guide provides a step-by-step approach to solving combinatorics problems using permutations and combinations formulas.

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