How many different permutations can be made from the letters in the word “CHICHARITO”?Question 6Answera.453,600b.1,814,400c.3,628,800d.907,200

1. Break Down the Problem

We need to calculate the number of different permutations of the letters in the word "CHICHARITO". The word has repeating letters, so we need to account for these repetitions in our calculation.

2. Relevant Concepts

The formula for calculating permutations of letters, taking into account repetitions, is given by:

$P = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}$

Where:

• $n$ is the total number of letters,
• $n_1, n_2, \ldots, n_k$ are the frequencies of the distinct letters.

3. Analysis and Detail

1. Count Total Letters:

   • The word "CHICHARITO" has 10 letters.

2. Count Frequencies of Each Letter:

   • C: 2
   • H: 2
   • I: 2
   • A: 1
   • R: 1
   • T: 1
   • O: 1

3. Substitute Values into the Formula:

   • Total letters $n = 10$
   • Frequencies: $n_C = 2$, $n_H = 2$, $n_I = 2$, $n_A = 1$, $n_R = 1$, $n_T = 1$, $n_O = 1$

   Plugging these into the formula gives us:

$P = \frac{10!}{2! \cdot 2! \cdot 2! \cdot 1! \cdot 1! \cdot 1! \cdot 1!}$

4. Verify and Summarize

1. Calculate the Factorials:

   • $10! = 3,628,800$
   • $2! = 2$, therefore $2! \cdot 2! \cdot 2! = 2 \times 2 \times 2 = 8$
   • $1! = 1$, hence these contribute $1$ each.

2. Final Calculation:

   • Substituting into our equation:

$P = \frac{3,628,800}{8} = 453,600$

Final Answer

The number of different permutations that can be made from the letters in the word “CHICHARITO” is 453,600, which corresponds to option a. 453,600.