Final answer:
The number of different ways that the letters of "difference" can be arranged is 151200. If the letters are mixed up in a random sequence, the probability that the letters will be in alphabetical order is 1/151200.
Step-by-step explanation:
Counting Arrangements of the Word 'DIFFERENCE'
To find the number of different ways that the letters of "difference" can be arranged, we need to count the permutations of the letters, considering that some letters are repeated. We have 10 letters in total: 1 'D', 2 'I's, 2 'F's, 3 'E's, 1 'R', and 1 'N'. The formula for permutations of letters with repetitions is given by:
n! / (n1! * n2! * ... * nk!),
where n is the total number of letters and ni is the number of times each distinct letter appears.
In this case, the formula becomes:
10! / (2! * 2! * 3!) = 3628800 / (2 * 2 * 6) = 151200.
There are 151200 unique arrangements of the letters in the word "difference".
Probability of Alphabetic Order
When the letters are arranged in a random sequence, there is only one way to arrange them in alphabetical order. The probability is therefore the number of favorable outcomes (1) divided by the total number of possible arrangements (151200). Hence, the probability is 1 / 151200.