list all unique permutations of A B C D E F that do not repeat any letters with permutations that have 1, 2, 3, 4, 5, and 6 letters

To list the unique permutations of the letters \( A, B, C, D, E, F \) without repeating any letters, we need to generate permutations for different subset sizes: 1, 2, 3, 4, 5, and 6 letters.

### 1. **Permutations with 1 letter:**
There are 6 letters to choose from. The set of unique permutations is:
\[
A, B, C, D, E, F
\]

### 2. **Permutations with 2 letters:**
For 2-letter permutations, there are \( P(6,2) = 6 \times 5 = 30 \) unique permutations. These are:
\[
AB, AC, AD, AE, AF, BA, BC, BD, BE, BF, CA, CB, CD, CE, CF, DA, DB, DC, DE, DF, EA, EB, EC, ED, EF, FA, FB, FC, FD, FE
\]

### 3. **Permutations with 3 letters:**
For 3-letter permutations, there are \( P(6,3) = 6 \times 5 \times 4 = 120 \) unique permutations. These include:
\[
ABC, ABD, ABE, ABF, ACB, ACD, ACE, ACF, ADB, ADC, ADE, ADF, AEB, AEC, AED, AEF, AFB, AFC, AFD, AFE, BAC, BAD, BAE, BAF, BCA, BCD, BCE, BCF, BDA, BDC, BDE, BDF, BEA, BEC, BED, BEF, BFA, BFC, BFD, BFE, CAB, CAD, CAE, CAF, CBA, CBD, CBE, CBF, CDA, CDB, CDE, CDF, CEA, CEB, CED, CEF, CFA, CFB, CFD, CFE, DAB, DAC, DAE, DAF, DBA, DBC, DBE, DBF, DCA, DCB, DCE, DCF, DEA, DEB, DEC, DEF, DFA, DFB, DFC, DFE, EAB, EAC, EAD, EAF, EBA, EBC, EBD, EBF, ECA, ECB, ECD, ECF, EDA, EDB, EDC, EDF, EFA, EFB, EFC, EFD, FAB, FAC, FAD, FAE, FBA, FBC, FBD, FBE, FCA, FCB, FCD, FCE, FDA, FDB, FDC, FDE, FEA, FEB, FEC, FED
\]
(Note: A total of 120 such permutations.)

### 4. **Permutations with 4 letters:**
For 4-letter permutations, there are \( P(6,4) = 6 \times 5 \times 4 \times 3 = 360 \) unique permutations.

### 5. **Permutations with 5 letters:**
For 5-letter permutations, there are \( P(6,5) = 6 \times 5 \times 4 \times 3 \times 2 = 720 \) unique permutations.

### 6. **Permutations with 6 letters:**
For 6-letter permutations, there is only 1 permutation sequence:
\[
P(6,6) = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\]
These include:
\[
ABCDEF, ABCDFE, ABCEDF, ABCEFD, ABCFDE, ABCFED, ABDCEF, ABDCFE, ABDECF, ABDEFC, ABDFCE, ABDFEC, ABEFCD, ABEFDC, ...
\]

There are too many permutations to list for lengths 4, 5, and 6, but the total number of permutations for each subset length is as follows:

- **1-letter**: \(6\) permutations
- **2-letter**: \(30\) permutations
- **3-letter**: \(120\) permutations
- **4-letter**: \(360\) permutations
- **5-letter**: \(720\) permutations
- **6-letter**: \(720\) permutations

Would you like me to go deeper into any of these permutations?

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### 5 Related Questions:
1. How do we compute permutations of a set mathematically?
2. What are the differences between permutations and combinations?
3. Can we calculate the number of permutations for other subset sizes, e.g., 7 letters?
4. How do permutations apply to real-world problems like scheduling?
5. What happens if we allow repetition in permutations?

### Tip:
For large sets, calculating the total number of permutations can be done quickly using factorials: \( P(n,r) = \frac{n!}{(n-r)!} \).

Explore the unique permutations of the letters A, B, C, D, E, F without repetition. This problem covers combinatorics and the permutation formula for subset sizes ranging from 1 to 6 letters, including detailed calculations for each set size. Suitable for high school students in Grades 10-12.

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