Questions: How many different ways can the letters of "success" be arranged? If the letters of "success" are arranged in a random order, what is the probability that the result will be "success"? The number of different ways that the letters of "success" can be arranged is 420. The probability that the random arrangement of letters will result in "success" is

Solution Steps

To solve this problem, we need to calculate the number of distinct permutations of the letters in the word "success" and then determine the probability that a random arrangement of these letters will result in the word "success".

1. Number of Distinct Permutations:

   • Calculate the factorial of the total number of letters.
   • Divide by the factorial of the number of occurrences of each repeated letter.

2. Probability Calculation:

   • The probability is the ratio of the number of favorable outcomes (which is 1, since there is only one "success") to the total number of distinct permutations.

To find the number of distinct permutations of the letters in "success", we use the formula for permutations of a multiset:

\[ \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} \]

where \( n \) is the total number of letters, and \( n_1, n_2, \ldots, n_k \) are the frequencies of each distinct letter.

For "success":

• Total letters, \( n = 7 \)
• Frequencies: \( s = 3 \), \( c = 2 \), \( u = 1 \), \( e = 1 \)

Thus, the number of distinct permutations is:

\[ \frac{7!}{3! \cdot 2! \cdot 1! \cdot 1!} = \frac{5040}{6 \cdot 2 \cdot 1 \cdot 1} = \frac{5040}{12} = 420 \]

The probability that a random arrangement of the letters will result in "success" is the ratio of the number of favorable outcomes to the total number of distinct permutations.

There is only one favorable outcome (the word "success" itself), so:

\[ \text{Probability} = \frac{1}{420} \approx 0.002381 \]

Final Answer