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Factorial calculations might sound a bit intimidating at first, but they are quite straightforward once you understand them. A factorial, denoted by an exclamation mark (!), refers to the product of an integer and all the positive integers below it. It's represented as \( n! \), where \( n \) is a non-negative integer. To put it simply, factorial of a number gives you the number of ways you can arrange that number of items.Here's how it works:

• \( 1! = 1 \)
• \( 2! = 1 \times 2 = 2 \)
• \( 3! = 1 \times 2 \times 3 = 6 \)
• \( 4! = 1 \times 2 \times 3 \times 4 = 24 \)

Calculating a factorial is just about multiplying all whole numbers from the given number down to 1. It's a key component in understanding permutations and combinations in math, which is why you encounter it when working with these kinds of problems.

Combinatorics is a fancy term for a field of mathematics focused on counting arrangements and combinations of objects. It's all about figuring out how many ways things can be selected or arranged.In the problem of finding how many ways the letters in the word "BANANA" can be arranged, we are tackling a part of combinatorics called 'permutations'. Specifically, these are permutations with repetitions.When letters or objects repeat, like the 'A's and 'N's in "BANANA", the formula for finding permutations must account for these repetitions. This ensures that identical items aren't counted multiple times incorrectly.The formula used is:\[P(n; n_1, n_2, ..., n_k) = \frac{n!}{n_1!n_2!...n_k!}\]This formula takes:

• \( n \) as the total number of objects
• \( n_1, n_2, ..., n_k \) as the frequencies of the repeating objects

Using this formula allows you to count distinct arrangements, which is central to solving this type of permutation problem.

Letter arrangements involve lining up letters in different sequences. The word "BANANA" provides a classic example of permutations with repetition. To solve this, we first count each letter's frequency, noticing that 'A' repeats thrice and 'N' repeats twice.Next, apply the formula for permutations with repetitions. Estimating unique arrangements involves dividing the factorial of the total number of letters by the factorial of each individual frequency:

• Overall letter count: 6 (from "BANANA")
• Frequency: B=1, A=3, N=2

The formula becomes:\[\frac{6!}{1!\times 3!\times 2!}\]Calculate the factorials:

• \(6! = 720\)
• \(3! = 6\)
• \(2! = 2\)

Finally, by substituting back:\[\frac{720}{12} = 60\]Therefore, "BANANA" can be uniquely arranged in 60 different ways, acknowledging repeated letters correctly and avoiding overcounting.