Solution We Are To Compute The Number Of Distinct Permutations Of The Letters In The Word Mathematics
So, I was talking to a friend the other day, and we stumbled upon the word "mathematics". My friend, being the math enthusiast that she is, started geeking out about the beauty of numbers and patterns. I, on the other hand, started thinking about the word itself - what if we were to rearrange the letters to form new words?
This got me curious, and I started wondering about the number of distinct permutations of the letters in the word "mathematics". I mean, think about it - if you have a word with repeating letters, it's not as simple as just shuffling them around. You have to take into account the fact that some letters are indistinguishable from one another.
The Problem
So, let's break it down - the word "mathematics" has 11 letters: m-a-t-h-e-m-a-t-i-c-s. Some of these letters appear only once, while others, like "m" and "a", appear multiple times. To find the number of distinct permutations, we need to use the formula for permutations with repetition - and that's where things get interesting.
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The Formula
The formula is pretty straightforward: we divide the total number of permutations (which is the factorial of the number of letters) by the factorials of the number of times each letter repeats. In this case, we have 2 m's, 2 a's, and the rest of the letters appear only once. So, the formula becomes: 11! / (2! * 2!).
Now, I know what you're thinking - what's the point of all this? Well, my curious friend, it's actually pretty cool to think about the number of hidden patterns in a word like "mathematics". And who knows, maybe one day we'll discover a new mathematical concept hidden in the letters themselves.
So, after doing the math (pun intended), we get: 11! / (2! * 2!) = 4989600. That's a lot of distinct permutations! I hope you're as amazed as I am by the sheer number of possibilities hidden in a single word. And who knows, maybe next time you see the word "mathematics", you'll think about all the secret codes waiting to be uncovered.
