Problem Description:
The image shows a "Free Printable Graduation Banner" that includes the entire alphabet and numbers. The task is to solve a problem related to this banner. Since no specific problem is stated in the image, I will infer a common type of problem associated with such banners:
How many different ways can you arrange the letters in the word "CONGRATULATIONS"?
---
Solution:
#### Step 1: Understand the Problem
We need to determine how many unique arrangements (permutations) of the letters in the word "CONGRATULATIONS" are possible.
#### Step 2: Count the Total Letters
The word "CONGRATULATIONS" has 14 letters.
#### Step 3: Identify Repeated Letters
To calculate the number of unique permutations, we must account for repeated letters. Here is the breakdown of the letters:
- C: 1
- O: 1
- N: 1
- G: 1
- R: 1
- A: 2
- T: 2
- U: 2
- L: 1
- I: 1
- O: 1
- N: 1
From this, we see that the letters
A,
T, and
U each appear twice.
#### Step 4: Use the Permutation Formula for Repeated Letters
The formula for the number of unique permutations of a word with repeated letters is:
\[
\text{Number of permutations} = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}
\]
Where:
- \( n \) is the total number of letters.
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeated letters.
For "CONGRATULATIONS":
- \( n = 14 \)
- \( n_A = 2 \), \( n_T = 2 \), \( n_U = 2 \)
Thus, the formula becomes:
\[
\text{Number of permutations} = \frac{14!}{2! \cdot 2! \cdot 2!}
\]
#### Step 5: Calculate Factorials
First, calculate \( 14! \):
\[
14! = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 87,178,291,200
\]
Next, calculate \( 2! \):
\[
2! = 2 \times 1 = 2
\]
Since there are three \( 2! \) terms in the denominator:
\[
2! \cdot 2! \cdot 2! = 2 \times 2 \times 2 = 8
\]
#### Step 6: Compute the Final Result
Now, divide \( 14! \) by \( 8 \):
\[
\frac{14!}{2! \cdot 2! \cdot 2!} = \frac{87,178,291,200}{8} = 10,897,286,400
\]
#### Final Answer:
The number of unique arrangements of the letters in "CONGRATULATIONS" is:
\[
\boxed{10897286400}
\]
Parent Tip: Review the logic above to help your child master the concept of free printable letters gold.
