The exact number of different ways to arrange the letters in the word 'PROBABILITY' is 9,979,200.
Understanding Letter Arrangements (Permutations)
Arranging letters to form different sequences is a concept known as permutations in mathematics. When dealing with permutations, two main factors are considered: the total number of items and whether any items are identical (repeated).
Total Letters in 'PROBABILITY'
The word 'PROBABILITY' consists of 11 letters.
Let's list them and count their occurrences:
| Letter | Count |
|---|---|
| P | 1 |
| R | 1 |
| O | 1 |
| B | 2 |
| A | 1 |
| I | 2 |
| L | 1 |
| T | 1 |
| Y | 1 |
| Total | 11 |
The Factorial Concept
To calculate the number of arrangements, we typically use the factorial function, denoted by an exclamation mark (!). n factorial (n!) is the product of all positive integers less than or equal to n.
For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
If all 11 letters in 'PROBABILITY' were unique (i.e., no letters were repeated), the number of possible arrangements would be simply 11!.
The calculation for 11! is:
11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
11! = 39,916,800
Accounting for Repeated Letters
However, the word 'PROBABILITY' has repeated letters, which reduces the number of unique arrangements. If we simply calculated 11!, we would be overcounting arrangements because swapping identical letters (e.g., the two 'B's) doesn't create a new, distinct arrangement.
To correct for these repetitions, we divide the total factorial by the factorial of the count of each repeated letter. The formula for permutations with repetitions is:
$$ \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} $$
Where:
- n is the total number of letters.
- n1, n2, ..., nk are the counts of each repeated letter.
In 'PROBABILITY':
- Total letters (n) = 11
- The letter 'B' appears 2 times (nB = 2)
- The letter 'I' appears 2 times (nI = 2)
Now, let's apply the formula:
-
Calculate the factorial of the total number of letters:
11! = 39,916,800 -
Calculate the factorials for the counts of repeated letters:
- For 'B': 2! = 2 × 1 = 2
- For 'I': 2! = 2 × 1 = 2
-
Divide the total factorial by the product of the factorials of the repeated letter counts:
$$ \frac{11!}{2! \cdot 2!} = \frac{39,916,800}{2 \cdot 2} = \frac{39,916,800}{4} $$9,979,200
Therefore, considering the repeated letters, there are 9,979,200 different ways to arrange the letters in the word 'PROBABILITY'.
