how many strings made of 3 letters is it possible to create from the letters SMALLEXAM?

M can be repeated twice in a string but not more
S can't be repeated more than once

possible answers: 680, 504, 135, no answer is correct

To determine the number of possible 3-letter strings from the letters SMALLEXAM, given the constraints, we need to systematically analyze the options.

### Letters Available:
- S, M, A, L, L, E, X, A, M

### Constraints:
1. M can be repeated at most twice.
2. S can be used at most once.

### Steps to Calculate the Possible Strings:

1. **Calculate combinations with 0 M's:**
   - Remaining letters: S, A, L, L, E, X, A (7 letters)
   - Possible combinations: \( \binom{7}{3} = 35 \)

2. **Calculate combinations with 1 M:**
   - Remaining letters: S, A, L, L, E, X, A (7 letters)
   - Choose 2 out of 7: \( \binom{7}{2} = 21 \)
   - Include M in each combination: 21 combinations

3. **Calculate combinations with 2 M's:**
   - Remaining letters: S, A, L, L, E, X, A (7 letters)
   - Choose 1 out of 7: \( \binom{7}{1} = 7 \)
   - Include MM in each combination: 7 combinations

4. **Sum of combinations:**
   - 0 M's: 35 combinations
   - 1 M: 21 combinations
   - 2 M's: 7 combinations
   - Total: \( 35 + 21 + 7 = 63 \)

However, we must consider combinations where S appears more than once (invalid due to constraints). We systematically consider valid strings:

### Valid Strings Analysis:

**1. 0 M's:**
   - \( \binom{7}{3} = 35 \) (excluding S repetitions)

**2. 1 M:**
   - M + \( \binom{7}{2} \) without exceeding 1 S = 21

**3. 2 M's:**
   - MM + \( \binom{7}{1} \) = 7

**Final Calculation with Constraints Applied:**

Correct possible combinations based on constraints:
\[ 35 + 21 + 7 = 63 \]

Since none of the given choices (680, 504, 135) fit, the correct answer is:
\[ \text{no answer is correct} \]

Do you want more details or have any questions? Here are some questions you might ask:

1. How to calculate combinations with constraints in general?
2. What are the steps to solve similar permutation problems?
3. Can you give an example with different constraints?
4. How do we account for constraints in probability?
5. Why are permutations different from combinations?

**Tip:** When solving problems with constraints, break down each condition and check for overlapping constraints to ensure accurate calculations.

Learn how to calculate the number of possible 3-letter strings from the letters SMALLEXAM, considering constraints on repetitions of S and M. This problem involves combinatorial analysis using the binomial coefficient formula. Suitable for high school students studying combinatorics and permutations.

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