How many possibilities with 3 numbers?

Understand the Problem

The question is asking for the number of possible combinations or arrangements that can be made with 3 numbers. To answer this, we need to define if the numbers can be repeated and if the order matters, since this will affect the total number of possibilities.

Answer

$27$

Steps to Solve

Identify if repetition is allowed and if order matters

First, we need to determine if the 3 numbers can be repeated and if the order of arrangement is important. Let's assume we are finding combinations where repetition is allowed and order matters.

Calculate the total combinations

If repetition is allowed (numbers can be chosen more than once) and order matters, the total number of arrangements can be calculated using the formula:

$$ n^r $$

Where:

( n ) is the total number of choices (in this case, 3 numbers),
( r ) is the number of places (which is also 3 in this case).

So, we have:

$$ 3^3 $$

Perform the calculation

Now, calculate ( 3^3 ):

$$ 3^3 = 27 $$

This means there are 27 different arrangements possible with 3 numbers when both repetition is allowed and order matters.

The total number of possible arrangements is $27$.

More Information

This result shows the power of combinations when you allow repetition and consider order. For example, if we had 3 distinct digits (like 1, 2, 3), the arrangements would be quite varied, leading to a larger total number than you might initially expect.

Tips

Forgetting to clarify whether repetition is allowed or if the order matters, which can lead to incorrect calculations.
Confusing combinations (where order does not matter) with permutations (where order does matter).

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