Question

how many different 10-letter words (real or imaginary) can be formed from the following letters?

w, z, k, o, s, a, z, c, t, i

ten-letter words (real or imaginary) can be formed with the given letters.
(type a whole number.)

Explanation:

Count the total number of letters and identify duplicates

Using the Permutations of Multisets knowledge point
The given multiset of letters is:

$$ W, Z, K, O, S, A, Z, C, T, I $$

The total number of letters is \(n = 10\).
The frequency of each unique letter is:

• \(W\): 1
• \(Z\): 2
• \(K\): 1
• \(O\): 1
• \(S\): 1
• \(A\): 1
• \(C\): 1
• \(T\): 1
• \(I\): 1

Apply the multiset permutation formula

Using the Permutations of Multisets knowledge point
The formula for the number of distinct permutations of \(n\) items with repeating elements is:

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

Substituting \(n = 10\) and the frequency of \(Z\) which is 2:

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

Calculate the final numerical value

Using the Permutations of Multisets knowledge point
Calculate the factorial values:

$$ 10! = 3,628,800 $$

$$ P = \frac{3,628,800}{2} = 1,814,400 $$

Answer:

<blank>1,814,400</blank> ten-letter words (real or imaginary) can be formed with the given letters.