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Permutations are a way to organize or arrange a set of items where order is important. In simple terms, permutations count how many different ways we can place items in a sequence. For example, if you have the word "BOOLEAN" and want to determine different ways to rearrange its letters, you would calculate its permutations.
The general formula for permutations of a set of items is the factorial of the number of items, denoted as "n!". In the word "BOOLEAN," since there are 7 letters, you might first think there are 7! ways to permute them.
However, permutations become slightly more complex when identical items are present. This is why understanding permutations is crucial when dealing with repeated elements.

When calculating permutations for a word with repeated letters, it's essential to account for repetitions to avoid overcounting equivalent arrangements. Here's the key: If you have any letter that repeats, say 'n' times, you divide the total number of permutations by the factorial of the number of times the letter appears. This corrects the calculation by removing duplicates caused by the repeated elements.
In "BOOLEAN," the letter 'O' repeats twice. Thus, instead of 7! permutations, we calculate \[ \frac{7!}{2!} \]
If we didn't adjust for the repeated 'O's, each arrangement where they switch places would seem distinct when actually they are the same.
Understanding how to adjust for repeated elements in permutations is vital in many combinatorial problems.

Combinatorial probability involves finding the likelihood of a specific arrangement or combination occurring among possible arrangements. It's all about dividing the number of favorable outcomes by the number of total possible outcomes.
In the case of the word "BOOLEAN" and the question of whether the two O's are together, we start by treating the pair of O's as a single unit. This conceptually reduces the size of the problem, allowing us to count fewer permutations, specifically \[ 6! \times 2! \].
The favorable permutations are the ones where the O's stick together.
The final step is to employ probability by dividing these favorable permutations by the total permutations we discussed earlier. This ratio determines the chance of your desired event occurring, providing a clear insight into the probability calculation.

• Numerator: Permutations with O's together.
• Denominator: Total permutations, corrected for repetition.

Applying combinatorial probability helps in decoding complex situations with a straightforward arithmetic solution.