1. **Problem statement:** Find the number of distinguishable permutations of the letters "AAABBBCC". 2. **Formula:** The number of distinguishable permutations of $n$ objects where there are groups of identical objects is given by: $$\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n$ is the total number of letters, and $n_1, n_2, \ldots, n_k$ are the counts of each identical letter. 3. **Apply to the problem:** - Total letters $n = 8$ (3 A's, 3 B's, 2 C's) - So the number of distinguishable permutations is: $$\frac{8!}{3! \times 3! \times 2!}$$ 4. **Calculate factorials:** - $8! = 40320$ - $3! = 6$ - $2! = 2$ 5. **Substitute and simplify:** $$\frac{40320}{6 \times 6 \times 2} = \frac{40320}{72}$$ 6. **Simplify with cancellation:** $$\frac{40320}{\cancel{72}} = \frac{40320}{\cancel{72}}$$ Calculate $40320 \div 72 = 560$ 7. **Final answer:** There are $560$ distinguishable permutations of the letters "AAABBBCC". --- 1. **Problem statement:** Find the probability that the first letter of a randomly chosen permutation is "A". 2. **Total permutations:** $560$ (from part a) 3. **Number of permutations with first letter "A":** Fix first letter as "A", then permute remaining letters "AABBBCC". - Remaining letters count: 7 letters with 2 A's, 3 B's, 2 C's - Number of such permutations: $$\frac{7!}{2! \times 3! \times 2!}$$ 4. **Calculate factorials:** - $7! = 5040$ - $2! = 2$ - $3! = 6$ 5. **Substitute and simplify:** $$\frac{5040}{2 \times 6 \times 2} = \frac{5040}{24} = 210$$ 6. **Probability:** $$P(\text{first letter is A}) = \frac{210}{560}$$ 7. **Simplify fraction:** $$\frac{\cancel{210}}{\cancel{560}} = \frac{3}{8}$$ 8. **Final answer:** The probability that the first letter is "A" is $\frac{3}{8}$. --- 1. **Problem statement:** Find the probability that after shuffling the cards with letters "AAABBBCC" and laying them out again, the arrangement is exactly the same as before. 2. **Total number of possible arrangements:** $560$ (from part a) 3. **Since all arrangements are equally likely, the probability of the same arrangement is:** $$\frac{1}{560}$$ 4. **Final answer:** The probability of ending up with exactly the same arrangement after shuffling is $\frac{1}{560}$.