Question 5
10 pts
How many. distinguishable permutations of the letters in the word WEWAHITCHKA are possible?

Step 1 ：The word 'WEWAHITCHKA' has 11 letters in total. However, there are some letters that are repeated. Specifically, the letter 'W' appears twice, the letter 'A' appears twice, and the letter 'H' appears twice. All other letters appear once.

Step 2 ：The number of distinguishable permutations of a set of objects, some of which are identical, is given by the formula: \[\frac{n!}{n1! * n2! * ... * nk!}\] where n is the total number of objects, and n1, n2, ..., nk are the numbers of each type of identical object.

Step 3 ：In this case, n = 11, n1 = n2 = n3 = 2 (for the letters 'W', 'A', and 'H'), and all other ni = 1.

Step 4 ：So, the number of distinguishable permutations is: \[\frac{11!}{2! * 2! * 2! * 1! * 1! * 1! * 1! * 1! * 1! * 1!}\]

Step 5 ：Calculating the above expression, we get the number of distinguishable permutations as 4989600.0

Step 6 ：Final Answer: The number of distinguishable permutations of the letters in the word 'WEWAHITCHKA' is \(\boxed{4989600}\)