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The problem asks how many ways words can be created using the letters \(U, S, C\) under specific conditions. Let's break down the problem step-by-step to solve it.

### Conditions:
1. **Each letter is used at least once**: So, the word must include at least one \(U\), one \(S\), and one \(C\).
2. **The total length is 6**: The word has exactly 6 letters.
3. **At least as many \(U\)'s as \(S\)'s**: The number of \(U\)'s must be greater than or equal to the number of \(S\)'s.
4. **At least as many \(S\)'s as \(C\)'s**: The number of \(S\)'s must be greater than or equal to the number of \(C\)'s.
5. **Lexicographically first word**: The word must be the lexicographically smallest, meaning the letters should appear in increasing order as per the alphabetical sequence \(C < S < U\).

### Step-by-Step Solution:
#### Total number of letters:
The word must contain exactly 6 letters, and since each letter is used at least once, let us define the number of each letter as:
- \(x_U\) = number of \(U\)'s,
- \(x_S\) = number of \(S\)'s,
- \(x_C\) = number of \(C\)'s.

The total length of the word gives the equation:
\[
x_U + x_S + x_C = 6
\]

#### Applying constraints:
From the conditions:
- \(x_U \geq x_S\)
- \(x_S \geq x_C\)
- Each letter must appear at least once, i.e., \(x_U, x_S, x_C \geq 1\).

#### Finding valid solutions:
Now, we solve the equation \(x_U + x_S + x_C = 6\) under these constraints:
- Since \(x_U \geq x_S \geq x_C\), the possible valid sets of \((x_U, x_S, x_C)\) are:
    1. \((4, 2, 1)\)
    2. \((3, 2, 1)\)
    3. \((3, 3, 1)\)
    4. \((2, 2, 2)\)

#### Lexicographically first word:
The lexicographically first arrangement of the word requires that the letters appear in alphabetical order \(C < S < U\). So, for each valid set of \((x_U, x_S, x_C)\), we need to count how many ways we can arrange the letters with the smallest sequence of letters coming first.

For each of these valid sets:
1. **For \((4, 2, 1)\)**: The word contains 4 \(U\)'s, 2 \(S\)'s, and 1 \(C\). The number of distinct ways to arrange these letters is:
   \[
   \frac{7!}{4!2!1!} = 105
   \]

2. **For \((3, 2, 1)\)**: The word contains 3 \(U\)'s, 2 \(S\)'s, and 1 \(C\). The number of distinct ways to arrange these letters is:
   \[
   \frac{6!}{3!2!1!} = 60
   \]

3. **For \((3, 3, 1)\)**: The word contains 3 \(U\)'s, 3 \(S\)'s, and 1 \(C\). The number of distinct ways to arrange these letters is:
   \[
   \frac{7!}{3!3!1!} = 140
   \]

4. **For \((2, 2, 2)\)**: The word contains 2 \(U\)'s, 2 \(S\)'s, and 2 \(C\)'s. The number of distinct ways to arrange these letters is:
   \[
   \frac{7!}{2!2!2!} =12
   


How many ways can you create words using the letters U, S, C where each letter is used at least once, the total length is 6, at least as many U's are used as S's, at least as many S's are used as C's, and the word is lexicographically first among all of its rearrangements.

Learn how to solve a combinatorics problem involving the letters U, S, C under specific constraints. The problem explores word creation with permutations, ensuring conditions on the usage of letters and lexicographical order. Step-by-step explanation with formulas and permutation theory. Suitable for undergraduate students.

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