Question

use your existing knowledge of casework, combinations, and permutations to answer the questions below. consider question a and question b, but do not solve them. then answer question 1. a. how many unique ways are there to arrange the letters in the word duck? b. how many unique ways are there to arrange the letters in the word tree? 1. will the answers to question a and question b be equal? explain why or why not.

Explanation:

To determine if the answers to A and B are equal, we analyze the letter arrangements:

• For "DUCK": All 4 letters (D, U, C, K) are distinct. The number of permutations of \( n \) distinct objects is \( n! \), so for \( n = 4 \), it's \( 4! = 24 \).
• For "TREE": The letters are T, R, E, E. Here, "E" is repeated twice. The formula for permutations of \( n \) objects with \( k \) identical objects is \( \frac{n!}{k!} \). For \( n = 4 \) and \( k = 2 \) (the two E's), it's \( \frac{4!}{2!} = \frac{24}{2} = 12 \).

Since \( 24
eq 12 \), the number of arrangements differs because "TREE" has repeated letters (affecting the permutation count) while "DUCK" has all distinct letters.

Answer:

No, the answers will not be equal. The word "DUCK" has 4 distinct letters, so its permutations are \( 4! = 24 \). The word "TREE" has 4 letters with "E" repeated twice, so its permutations are \( \frac{4!}{2!} = 12 \). The repetition of "E" in "TREE" reduces the number of unique arrangements compared to "DUCK" (with all distinct letters).