# How many 4 letter combinations are there?

#### Understand the Problem

The question is asking for the total number of unique combinations that can be formed using 4 letters. This involves calculating permutations or combinations based on the context, such as whether letters can repeat or if they must be unique.

$14950$

The total number of unique combinations of 4 letters from the alphabet is $14950$.

#### Steps to Solve

1. Determine the scenario for combinations
Since the question mentions "unique combinations," we will assume that the letters cannot repeat. Therefore, we will focus on combinations.

2. Identify the total number of letters
If we are using the English alphabet, there are 26 letters available to choose from.

3. Calculate combinations using the formula
The formula for combinations is given by:
$$C(n, r) = \frac{n!}{r!(n - r)!}$$
Where ( n ) is the total number of items (letters) to choose from, and ( r ) is the number of items to choose (4 in this case).

4. Plug in the numbers
In this case, we have:

• ( n = 26 ) (total letters)
• ( r = 4 ) (letters to choose)
So we need to calculate:
$$C(26, 4) = \frac{26!}{4!(26 - 4)!}$$
This simplifies to:
$$C(26, 4) = \frac{26!}{4! \cdot 22!}$$
1. Calculate the factorials
We will calculate ( 26! ) divided by ( 22! ):
$$C(26, 4) = \frac{26 \times 25 \times 24 \times 23}{4!}$$
And since ( 4! = 24 ), we get:
$$C(26, 4) = \frac{26 \times 25 \times 24 \times 23}{24}$$

2. Final Calculation
Now, cancel out the 24:
$$C(26, 4) = 26 \times 25 \times 23$$
Calculate this to get the total number of unique combinations.

The total number of unique combinations of 4 letters from the alphabet is $14950$.

You can create a variety of combinations using groups of letters, which can be useful in password generation, bingo games, or any scenario that requires selection. In combinatorial problems, understanding the difference between permutations and combinations is crucial for accurate calculations.

#### Tips

• Forgetting the difference between combinations and permutations. Make sure to identify whether the order matters (permutations) or not (combinations).
• Miscalculating factorial functions, especially for larger numbers. It helps to simplify before calculating large factorials.
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