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Step-by-Step Solution
Step 1: Identify the Distinct Letters and Their Frequencies
The given word “DISTRIBUTION” contains the following letters with their respective counts:
D (1)
I (3)
S (1)
T (2)
R (1)
B (1)
U (1)
O (1)
N (1)
So we have three I’s, two T’s, and each of the other letters appearing exactly once.
Step 2: Classify the Possible Types of 4-Letter Selections
We aim to form 4-letter arrangements from these letters. The scenarios based on possible repetitions are:
3 identical letters + 1 distinct letter ( a, a, a, b )
2 identical letters + 2 identical letters ( a, a, b, b )
2 identical letters + 1 distinct + 1 distinct ( a, a, b, c )
4 distinct letters ( a, b, c, d )
Step 3: Calculate the Number of Arrangements for Each Scenario
3 identical + 1 distinct ( a, a, a, b )
Only the letter I can appear three times. The distinct letter can be any of the other 8 letter types (D, S, T, R, B, U, O, N).
Number of ways to choose the distinct letter: \binom{8}{1} = 8 .
Number of ways to arrange 3 identical letters and 1 distinct: \frac{4!}{3!} = 4 .
Total arrangements in this scenario: 8 \times 4 = 32 .
2 identical letters + 2 identical letters ( a, a, b, b )
The only possibility is having 2 I’s and 2 T’s.
Number of ways to arrange I, I, T, T: \frac{4!}{2!\,2!} = 6 .
Total arrangements in this scenario: 6 .
2 identical + 1 distinct + 1 distinct ( a, a, b, c )
We can choose either I or T to be the letter that appears twice ( 2 ways). Then we choose any 2 distinct letters out of the 8 remaining single-frequency letters ( \binom{8}{2} = 28 ways).
Number of ways to arrange these 4 letters (with 2 identical and 2 distinct): \frac{4!}{2!} = 12 .
Total arrangements in this scenario: 2 \times 28 \times 12 = 672 .
4 distinct letters ( a, b, c, d )
There are 9 distinct letters in total (D, I, S, T, R, B, U, O, N). We choose any 4 of these 9 letters and arrange them.
Number of ways to choose 4 distinct letters: \binom{9}{4} = 126 .
Number of ways to arrange these 4 letters: 4! = 24 .
Total arrangements in this scenario: 126 \times 24 = 3024 .
Step 4: Sum All Possible Arrangements
Add the results from all four scenarios:
32 + 6 + 672 + 3024 = 3734.
Final Answer
The total number of 4-letter arrangements that can be formed from “DISTRIBUTION” is 3734.
