Question

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question 7 (3 points)
how many ways can you arrange the letters in the word possibility?

Explanation:

Step1: Count total and repeated letters

First, count the total number of letters in "POSSIBILITY" and the frequency of each repeated letter. The word "POSSIBILITY" has 12 letters. Let's list the frequency of each letter:

• P: 1
• O: 1
• S: 3 (since there are three S's)
• I: 2 (two I's)
• B: 1
• L: 2 (two L's)
• T: 1
• Y: 1

The formula for permutations of a word with repeated elements is $\frac{n!}{n_1! \times n_2! \times \dots \times n_k!}$, where $n$ is the total number of elements, and $n_1, n_2, \dots, n_k$ are the frequencies of the repeated elements.

Here, $n = 12$, $n_S = 3$, $n_I = 2$, $n_L = 2$.

Step2: Calculate factorials

Calculate $12!$, $3!$, $2!$, and $2!$.

• $12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479001600$
• $3! = 3 \times 2 \times 1 = 6$
• $2! = 2 \times 1 = 2$ (for both I and L, so we have $2! \times 2!$)

Step3: Apply the formula

Now, apply the formula:
$\frac{12!}{3! \times 2! \times 2!} = \frac{479001600}{6 \times 2 \times 2}$

First, calculate the denominator: $6 \times 2 \times 2 = 24$

Then, divide the numerator by the denominator: $\frac{479001600}{24} = 19958400$

Answer:

19958400