1. **State the problem:** We want to find how many four-letter words can be formed from the letters of the word ATTENTIVENESS, and how many of these words have at least one vowel. 2. **Analyze the letters:** The word ATTENTIVENESS has the letters: A, T, T, E, N, T, I, V, E, N, E, S, S. Count each letter's frequency: - A: 1 - T: 3 - E: 3 - N: 2 - I: 1 - V: 1 - S: 2 3. **Total letters available:** 13 letters with repetitions as above. 4. **Step 1: Calculate total number of 4-letter words (with repetition limits):** We need to count the number of distinct 4-letter arrangements from these letters respecting the maximum counts. This is a complex problem involving counting permutations with limited repetitions. 5. **Approach:** We consider all possible multisets of letters of size 4 that can be formed from the letters with their max counts, then count permutations for each multiset. 6. **Step 2: Calculate number of 4-letter words with no vowels:** Vowels are A, E, I. Consonants are T, N, V, S. Consonant counts: - T: 3 - N: 2 - V: 1 - S: 2 We find number of 4-letter words formed only from consonants. 7. **Step 3: Use complementary counting:** Number of 4-letter words with at least one vowel = Total 4-letter words - Number of 4-letter words with no vowels. 8. **Calculate total 4-letter words:** We enumerate all possible letter count patterns for 4 letters considering max counts. Possible patterns for letter counts in 4-letter word: - 4 distinct letters - 1 letter repeated twice + 2 distinct letters - 2 letters repeated twice - 1 letter repeated thrice + 1 distinct letter - 1 letter repeated 4 times (not possible here since max repetition is 3) 9. **Calculate number of multisets and permutations for each pattern:** - Pattern A: 4 distinct letters Choose 4 distinct letters from 7 distinct letters (A,T,E,N,I,V,S) = $\binom{7}{4} = 35$ Each arrangement has $4! = 24$ permutations Total = $35 \times 24 = 840$ - Pattern B: 1 letter repeated twice + 2 distinct letters Choose letter to repeat twice (must have at least 2 copies): T(3), E(3), N(2), S(2) → 4 choices Choose 2 distinct letters from remaining 6 letters (excluding chosen repeated letter) = $\binom{6}{2} = 15$ Number of permutations for multiset with counts (2,1,1) = $\frac{4!}{2!} = 12$ Total = $4 \times 15 \times 12 = 720$ - Pattern C: 2 letters repeated twice Choose 2 letters with at least 2 copies from T,E,N,S (4 letters) = $\binom{4}{2} = 6$ Number of permutations for multiset (2,2) = $\frac{4!}{2!2!} = 6$ Total = $6 \times 6 = 36$ - Pattern D: 1 letter repeated thrice + 1 distinct letter Letters with at least 3 copies: T(3), E(3) Choose 1 letter from these 2 for triple Choose 1 distinct letter from remaining 6 letters = 6 Number of permutations for multiset (3,1) = $\frac{4!}{3!} = 4$ Total = $2 \times 6 \times 4 = 48$ - Pattern E: 1 letter repeated 4 times No letter has 4 copies, so 0 10. **Sum total 4-letter words:** $$840 + 720 + 36 + 48 = 1644$$ 11. **Calculate 4-letter words with no vowels:** Consonants: T(3), N(2), V(1), S(2) Distinct consonants = 4 Repeat steps 8-9 for consonants only: - Pattern A: 4 distinct consonants Choose 4 distinct consonants from 4 = 1 Permutations = $4! = 24$ Total = 24 - Pattern B: 1 letter repeated twice + 2 distinct letters Letters with at least 2 copies: T(3), N(2), S(2) → 3 choices Choose 2 distinct letters from remaining 3 consonants (excluding repeated letter) = $\binom{3}{2} = 3$ Permutations = 12 Total = $3 \times 3 \times 12 = 108$ - Pattern C: 2 letters repeated twice Choose 2 letters with at least 2 copies from T,N,S (3 letters) = $\binom{3}{2} = 3$ Permutations = 6 Total = $3 \times 6 = 18$ - Pattern D: 1 letter repeated thrice + 1 distinct letter Letters with at least 3 copies: T(3) Choose 1 letter for triple = 1 Choose 1 distinct letter from remaining 3 consonants = 3 Permutations = 4 Total = $1 \times 3 \times 4 = 12$ - Pattern E: 1 letter repeated 4 times No letter with 4 copies 12. **Sum total no vowel words:** $$24 + 108 + 18 + 12 = 162$$ 13. **Calculate words with at least one vowel:** $$1644 - 162 = 1482$$ **Final answers:** - Total 4-letter words: $1644$ - 4-letter words with at least one vowel: $1482$