What do you get when arranging the letters of the word 'BALLOON' considering the repetitions?

Steps to Solve

1. Identify the total number of letters

In the word 'BALLOON', there are 7 letters in total: B, A, L, L, O, O, N.

2. Count the frequency of each distinct letter

Now we count how many times each letter appears:

• B: 1
• A: 1
• L: 2
• O: 2
• N: 1

3. Total distinct letters in the formula

We can summarize the letter counts:

• Total letters (n) = 7
• Frequencies: B = 1, A = 1, L = 2, O = 2, N = 1

4. Apply the permutation of multiset formula

Now, we can use the formula for permutations of a multiset:

$$ \text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdot n_4! \cdot n_5!} $$

Substituting in our values:

$$ \text{Number of arrangements} = \frac{7!}{1! \cdot 1! \cdot 2! \cdot 2! \cdot 1!} $$

5. Calculate the factorial values

Calculate the factorial values:

• $7! = 5040$
• $1! = 1$
• $2! = 2$

Now substitute these values into the formula:

$$ \text{Number of arrangements} = \frac{5040}{1 \cdot 1 \cdot 2 \cdot 2 \cdot 1} = \frac{5040}{4} = 1260 $$