In how many different ways can the 5 Olympic rings be colored Black, Red, Green, Yellow, and Blue, with one color for each ring and without changing the arrangement of the rings themselves?

Answer

To solve the problem of how many different ways the 5 Olympic rings can be colored using the colors Black, Red, Green, Yellow, and Blue, we can follow these steps:

### Step 1: Identify the number of rings and colors
We have 5 Olympic rings and 5 different colors available for coloring them.

### Step 2: Determine the choices for each ring
- For the **first ring**, we have **5 choices** (any of the 5 colors).
- For the **second ring**, since one color has already been used for the first ring, we have **4 choices** left.
- For the **third ring**, we have **3 choices** left after coloring the first two rings.
- For the **fourth ring**, we will have **2 choices** remaining.
- Finally, for the **fifth ring**, there will be only **1 choice** left.

### Step 3: Calculate the total arrangements
To find the total number of ways to color the rings, we multiply the number of choices for each ring:

\[ \text{Total arrangements} = 5 \times 4 \times 3 \times 2 \times 1 = 5! \]

### Step 4: Compute the factorial
Calculating \(5!\):

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Conclusion
Thus, the total number of different ways to color the 5 Olympic rings is **120**.

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