To solve these probability questions, we need to determine the total number of possible arrangements of the letters in the word "CALCULATE" and then find the number of favorable outcomes for each specific condition. The probability is the ratio of favorable outcomes to the total number of arrangements.
- Probability that the first letter is U: Count the number of arrangements where U is the first letter and divide by the total number of arrangements.
- Probability that the first letter is C: Count the number of arrangements where C is the first letter and divide by the total number of arrangements.
- Probability that the first letter is a vowel: Count the number of arrangements where the first letter is a vowel (A, U, E) and divide by the total number of arrangements.
The word "CALCULATE" consists of 9 letters, with the following counts for each letter:
- C: 2
- A: 2
- L: 2
- U: 1
- T: 1
- E: 1
The total number of distinct arrangements of the letters is calculated using the formula for permutations of multiset:
\[
\text{Total Arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots}
\]
Substituting the values, we have:
\[
\text{Total Arrangements} = \frac{9!}{2! \cdot 2! \cdot 2! \cdot 1! \cdot 1! \cdot 1!} = 45360
\]
To find the probability that the first letter is U, we calculate the number of arrangements where U is the first letter. The remaining letters are C, A, L, C, L, A, T, E (8 letters with counts: C: 2, A: 2, L: 2, T: 1, E: 1).
The number of favorable arrangements is:
\[
\text{Favorable}_U = \frac{8!}{2! \cdot 2! \cdot 2! \cdot 1! \cdot 1!} = 5040
\]
Thus, the probability is:
\[
P(U) = \frac{\text{Favorable}_U}{\text{Total Arrangements}} = \frac{5040}{45360} \approx 0.111
\]
Similarly, for the first letter being C, the remaining letters are A, L, C, U, L, A, T, E (8 letters with counts: A: 2, C: 1, L: 2, U: 1, T: 1, E: 1).
The number of favorable arrangements is:
\[
\text{Favorable}_C = \frac{8!}{1! \cdot 2! \cdot 2! \cdot 1! \cdot 1! \cdot 1!} = 10080
\]
Thus, the probability is:
\[
P(C) = \frac{\text{Favorable}_C}{\text{Total Arrangements}} = \frac{10080}{45360} \approx 0.222
\]
The vowels in "CALCULATE" are A, U, and E. We calculate the favorable arrangements for each vowel being the first letter.
For A: Remaining letters are C, L, C, U, L, A, T, E (8 letters).
\[
\text{Favorable}_A = \frac{8!}{1! \cdot 2! \cdot 2! \cdot 1! \cdot 1! \cdot 1!} = 10080
\]
For U: Already calculated as \( \text{Favorable}_U = 5040 \).
For E: Remaining letters are C, A, L, C, U, L, A, T (8 letters).
\[
\text{Favorable}_E = \frac{8!}{2! \cdot 2! \cdot 2! \cdot 1! \cdot 1!} = 5040
\]
Adding these favorable outcomes gives:
\[
\text{Favorable}_{\text{vowels}} = \text{Favorable}_A + \text{Favorable}_U + \text{Favorable}_E = 10080 + 5040 + 5040 = 20160
\]
Thus, the probability is:
\[
P(\text{vowel}) = \frac{\text{Favorable}_{\text{vowels}}}{\text{Total Arrangements}} = \frac{20160}{45360} \approx 0.444
\]
The probabilities are:
- Probability that the first letter is U: \( \approx 0.111 \)
- Probability that the first letter is C: \( \approx 0.222 \)
- Probability that the first letter is a vowel: \( \approx 0.444 \)
Thus, the final answers are:
\[
\boxed{P(U) \approx 0.111, \quad P(C) \approx 0.222, \quad P(\text{vowel}) \approx 0.444}
\]
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