Find the number of distinguishable permutations of the given letters "AABBCC". Your answer is :
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
![**Question:**
Find the number of distinguishable permutations of the given letters "AABBCC".
**Answer input field:**
Your answer is: [_______________]
**Explanation:**
To determine the number of distinguishable permutations of the letters "AABBCC", you can use the formula for permutations of multiset:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
where \( n \) is the total number of letters, and \( n_1, n_2, \ldots, n_k \) are the frequencies of each distinct letter.
For "AABBCC":
- Total letters, \( n = 6 \)
- Frequency of A, \( n_1 = 2 \)
- Frequency of B, \( n_2 = 2 \)
- Frequency of C, \( n_3 = 2 \)
Substituting into the formula:
\[ \frac{6!}{2! \times 2! \times 2!} = \frac{720}{8} = 90 \]
**Result:**
The number of distinguishable permutations of the letters "AABBCC" is 90.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb788abcd-eef5-4f27-84e3-02815d880ffa%2Fcc836d73-48c8-4142-9b66-5bfcbef55124%2Faclnk9_processed.png&w=3840&q=75)
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